Congress has the power to issue subpoenas, and it does not go through the courts. That power is extremely broad and courts generally resist attempts to interfere with Congress's constitutional power under the Speech or Debate Clause – "for any Speech or Debate in either House, they shall not be questioned in any other Place" – (which the courts interpret as including inquiries and subpoena powers enabling such inquiries, Eastland v. United States Servicemen's Fund, 421 U.S. 491), but may intervene in the case of extreme violation of constitutional rights (McSurely v. McClellan, 521 F.2d 1024). As stated in Exxon Corp. v. FTC, 589 F.2d 582,
the courts will intervene to protect constitutional rights from
infringement by Congress, including its committees and members...
where constitutional rights are not violated, there is no warrant for
the judiciary to interfere with the internal procedures of Congress
There does not seem to be any overriding federal law pertaining to the Bureau of Prisons and Congressional subpoenas which says that the authorities must respond in a certain way to a Congressional subpoena, forcing authorities to resist or comply in a certain way. The only issue would be whether the subpoena is lawful, and given the breadth of Congressional subpoena power, it is extremely unlikely that such a subpoena would not be held by the courts to be legal. Congress certainly could take into consideration the option of delaying a hearing for a few years, and there is zero chance that they actually would do so. In other words, the authorities must comply with the subpoena.
If the authorities do not comply with the subpoena, the responsible individual can be the subject of a contempt resolution. If passed, then the sergeant-at-arms of either house can arrest the violator and he can theoretically be held in the Capitol jail for the remainder of the legislative session. Or, the Senate can, under its rules, seek a court order forcing compliance. Or, the presiding officer of the House or Senate can refer the matter to the District Attorney for DC, for prosecution as criminal contempt under 2 USC 192.