Speech or Debate Immunity and defensive litigation

Donald Trump has sued to stop compliance with 1/6 Committee subpoenas; defendants are the committee, chair Bennie Thompson, the Archives, and David Ferriero, the national archivist. The suit is the usual Trump bullshit, with allegations that the committee is "attempting to damage the republic itself" (as opposed to 1/6 itself, which has been converted into an act of patriotic heroism).

Regardless of the validity of the subpoena, I cannot figure out how a lawsuit can be brought or proceed against a member of Congress and a congressional committee, both of whom enjoy immunity from being "questioned in any other Place" for any speech or debate, which includes issuing legislative subpoenas. This offers another example of litigation being pushed into a defensive rather than offensive posture. The target of the subpoena cannot go on the offensive to enjoin enforcement; he is supposed to "stand on his privilege and go into contempt," challenging the validity of the subpoena as a defense in either a civil-enforcement proceeding or a criminal-contempt proceeding.