The best course of action for Barr would be to file an objection to the subpoena in the proper court (probably the U.S. District Court for the District of Columbia) under or by analogy to Federal Rule of Civil Procedure 45(d) (which governs disputes over subpoenas in civil cases in the federal courts), if he thinks that there are matters he cannot lawfully divulge or thinks it is improper to divulge even if they are not privileged, and to let a judge decide.
Generally speaking, in a civil action, you can only object to a subpoena by following this process and Congress has increasingly used the civil lawsuit discovery process as a benchmark. Failure to file an objection with the proper court, or at least in a response to Congress by the date required in the subpoena to produce the materials, will generally constitute a waiver of the objections that might otherwise have been asserted to the subpoena.
Simply not responding without explanation by the due date would be a pretty clear case of Contempt of Congress which is the basis for both a civil action and criminal contempt charges.
For example, in the Lynch case, cited below, the Court stated with respect to information that was withheld without any claim of privilege:
Failure to provide any grounds for withholding particular records does
not comply with the order or enable the Court to resolve defendant’s
privilege claims as to those documents. Accordingly, defendant must
produce the material withheld without any proffered justification.
This said, in any dispute between Congress and the Executive Branch there is always some uncertainty, and the courts strongly favor negotiation and conferral between the parties before bringing these matters to a head.
A general discussion of Congressional subpoenas can be found here, recognizing, however, that while there are a variety of grounds for redaction asserted in the Mueller report case, "Executive Privilege" is not among them and so the special considerations that apply to an assertion of executive privilege do not apply. The authority of the judicial branch to resolve these issues has been upheld, for example, in the cases of United States v. Nixon (U.S. 1974) and Committee on Oversight and Government Reform v. Lynch (D. D.C 2016) (both of which involved the more difficult scenario of an assertion of executive privilege in addition to the more ordinary assertions of privileges like the grand jury privilege).
Not infrequently, the judge will review the unredacted material in camera (i.e. privately in chambers without showing it to the requesting party) to determine if the claim of privilege or other basis for redaction is really valid (e.g. maybe something that was redacted under the label grand jury testimony is not, in fact, grand jury testimony). But, there is case law to support the notion that Congress would have to demonstrate some specific reason why it doubts the accuracy of the assertions of the executive branch regarding redactions in this particular case to make it necessary for there to be an in camera review. In the Lynch case (which is a non-precedential opinion itself) the Court said:
As for whether the redactions are what they purport to be, the Court
notes that counsel for even the most disputatious parties are often
called upon to trust each other, and that the judiciary relies
regularly on declarations by the executive branch that matters
redacted from FOIA productions are what they are described to be in
the Vaughn index. See Loving v. U.S. Dep’t of Def., 550 F.3d 32, 41
(D.C. Cir. 2008) (holding that district court had not abused its
discretion by relying on agency’s Vaughn index and declaration in
determining whether a disputed document contained segregable
portions); Judicial Watch, Inc. v. Consumer Fin. Prot. Bureau, 60 F.
Supp. 3d 1, 13 (D.D.C. 2014) (“The reviewing court may rely on the
description of the withheld records set forth in the Vaughn index and
the agency’s declaration that it released all segregable
information.”). The Court has been provided with no reason to believe
that its assistance is needed to verify for counsel for one branch of
government assertions made in pleadings by an officer of the court
representing another, equal branch of government. If in the end, a
neutral is required to read each individual redaction and confirm that
what the Department claims is simply a name or a telephone number is
in fact a name or a telephone number, the parties can arrange for that
on their own.
These discretionary issues are likely to be influenced by the partisan leanings of the particular judges involved.
Another question is to whom a subpoena could be directed. While attorney-general Barr is one possible person to whom it could be directed, Mueller himself is another possible person to whom a subpoena could be directed and that might lead to a more tractable counter-party in the lawsuit and might simplify some of the conflicts of interest present in a subpoena of the attorney-general himself that in criminal contempt cases is enforceable by his subordinates, i.e. U.S. attorneys, who are required by law to bring such charges.
