Are there any 'bright lines' that prevent an administration (not just the current administration, but future administrations as well) from regularly resorting to the national security argument when it can not find congressional support for its foreign policy initiatives?
Presidential power of that type might arise from congressional authorization, where Congress authorizes action when POTUS deems that such-and-such is the case. An example is 8 USC 1182(f), which says
Whenever the President finds that the entry of any aliens or of any class of aliens into the United States would be detrimental to the interests of the United States, he may by proclamation, and for such period as he shall deem necessary, suspend the entry of all aliens or any class of aliens as immigrants or nonimmigrants, or impose on the entry of aliens any restrictions he may deem to be appropriate.
Explicit constitutional powers are not numerous, but include commanding the armed forces and overseeing execution of national law by appointing and removing executive officers. POTUS is constitutionally the federal official responsible for relations with foreign nations, so he can make treaties (subject to Senate approval) and executive agreements (not subject to approval), can appoint US ambassadors and can receive foreign ambassadors. Executive agreements have the greatest potential for being an avenue for POTUS to act contrary to Congress, but they are limited to agreements pursuant to legislation, treaty and constitutional authority (and, as agreements, are not unilateral).
The constitutional powers of the President as Commander-in-Chief to introduce United States Armed Forces into hostilities, or into situations where imminent involvement in hostilities is clearly indicated by the circumstances, are exercised only pursuant to (1) a declaration of war, (2) specific statutory authorization, or (3) a national emergency created by attack upon the United States, its territories or possessions, or its armed forces
The law limits the length of engagement of the military, without Congressional approval, but there is no toothful restriction on deploying the military for up to 60 days.
50 USC Ch. 34 more generally addressed "national emergencies", and started by declaring an end to all existing declared emergencies. Other sections of the law require Congress to be notified and emergencies to be published in the Federal Register. An emergency can be terminated by POTUS or by act of Congress, and after 6 months (and every 6 months thereafter), Congress shall meet to discuss the emergency. POTUS must also annually re-declare the emergency to keep it in force.
These are procedural requirements, not content requirements. In general, you wold have to go through Title 50 to check for particular powers granted by Congress
The International Emergency Economic Powers Act grants powers to POTUS, with the following limit:
(a) Any authority granted to the President by section 1702 of this title may be exercised to deal with any unusual and extraordinary threat, which has its source in whole or substantial part outside the United States, to the national security, foreign policy, or economy of the United States, if the President declares a national emergency with respect to such threat.
(b) The authorities granted to the President by section 1702 of this title may only be exercised to deal with an unusual and extraordinary threat with respect to which a national emergency has been declared for purposes of this chapter and may not be exercised for any other purpose. Any exercise of such authorities to deal with any new threat shall be based on a new declaration of national emergency which must be with respect to such threat.
which appears more limiting than the War Powers Act. There is an extensive list of powers granted under 50 USC 1702, which in that domain could be called "bright" lines. Generally, the lines are only as bright as Congress has painted them.
The question conflates two issues that only sometimes overlap - emergency powers on one hand, and the authority of the President v. Congress on issues of foreign affairs and national security on the other.
Some of the big cases involving the emergency powers of the President are entirely domestic: FDR's bank holidays, a case involving the President intervening in a strike by a domestic labor union, and Presidential involvement in enforcing a desegregation order with the National Guard in Little Rock, Arkansas.
Some of the big cases involving the foreign policy and national security powers of the President are not emergencies, like the issue of where to located the U.S. embassy in Israel.
Sometimes the two issues overlap, but doctrinally, this isn't necessarily or even often the case.
The only emergency powers expressly in the constitution are those applicable to invasion, insurrection and failure to have a "republican" government. Other emergency powers are solely authorized by statute of which there are several, some of the more important of which are identified in the answer by @user6726
Many of the statutory emergency powers of the President apply to domestic emergencies like national disasters or shortages of supplies of certain kinds of pharmaceuticals.
Of course, statutory emergency powers of a President can be taken away by Congress if the President doesn't veto the law doing so, or Congress overrides that veto.
There is a parallel, not necessarily emergency related, line of jurisprudential debate over the extent to which national security and foreign policy power is vested exclusively in the President, or is subject to Congressional direction.
A good example of a case of the location of the U.S. embassy to Israel, which was the subject of legislation that several Presidents took the position was unconstitutional because it interfered with the foreign policy powers of the President (relocating an embassy after attempts since 1995 to do so is not an "emergency" in the usual sense of the word and has not been framed as one).
The argument that the President has sole authority over military and foreign affairs is complicated by the fact that there are many specific grants of authority to Congress (mostly in Article I, Section 8 of the United States Constitution) that relate to foreign policy to some extent. I note the provisions with foreign policy implications in bold:
The Congress shall have Power To lay and collect Taxes, Duties, Imposts and Excises, to pay the Debts and provide for the common Defence and general Welfare of the United States; but all Duties, Imposts and Excises shall be uniform throughout the United States;
To borrow money on the credit of the United States;
To regulate Commerce with foreign Nations, and among the several States, and with the Indian Tribes;
To establish an uniform Rule of Naturalization, and uniform Laws on the subject of Bankruptcies throughout the United States;
To coin Money, regulate the Value thereof, and of foreign Coin, and fix the Standard of Weights and Measures;
To provide for the Punishment of counterfeiting the Securities and current Coin of the United States;
To establish Post Offices and Post Roads;
To promote the Progress of Science and useful Arts, by securing for limited Times to Authors and Inventors the exclusive Right to their respective Writings and Discoveries;
To constitute Tribunals inferior to the supreme Court;
To define and punish Piracies and Felonies committed on the high Seas, and Offenses against the Law of Nations;
To declare War, grant Letters of Marque and Reprisal, and make Rules concerning Captures on Land and Water;
To raise and support Armies, but no Appropriation of Money to that Use shall be for a longer Term than two Years;
To provide and maintain a Navy;
To make Rules for the Government and Regulation of the land and naval Forces;
To provide for calling forth the Militia to execute the Laws of the Union, suppress Insurrections and repel Invasions;
To provide for organizing, arming, and disciplining, the Militia, and for governing such Part of them as may be employed in the Service of the United States, reserving to the States respectively, the Appointment of the Officers, and the Authority of training the Militia according to the discipline prescribed by Congress;
To exercise exclusive Legislation in all Cases whatsoever, over such District (not exceeding ten Miles square) as may, by Cession of particular States, and the acceptance of Congress, become the Seat of the Government of the United States, and to exercise like Authority over all Places purchased by the Consent of the Legislature of the State in which the Same shall be, for the Erection of Forts, Magazines, Arsenals, dock-Yards, and other needful Buildings; And
To make all Laws which shall be necessary and proper for carrying into Execution the foregoing Powers, and all other Powers vested by this Constitution in the Government of the United States, or in any Department or Officer thereof.
Particularly contentious is the extent to which Congress can direct military affairs and foreign policy indirectly through strings attached to its appropriations power, as opposed to directly.
The President's foreign policy power is also checked by the requirement that Treaties be approved by the U.S. Senate, but mitigated by the fact that other forms of binding international agreements (e.g. executive agreements) entered into by a President are recognized as having legal force.
Liberal judges have tended to read the national defense and foreign policy power of the President vis-a-vis Congress narrowly, while conservative judges have tended to read the national defense and foreign policy power of the President vis-a-vis Congress more broadly. One particularly contentious aspect of this is the "unitary executive theory", popular with conservatives, which argues that Congress has limited power to micromanage the internal workings of the executive branch.