They have two choices;
1> Throw people in jail until they comply with congressional subpoenas, or
2> Stop claiming that Congress has subpoena power, because any power you won't enforce doesn't functionally exist; if Congress demands you give them something, say "fuck you" and ignore them, because they can't do shit.
The Republicans are playing legislative chicken, expecting that the Democrats won't actually enforce the law. But that's pushing it to the point where either they act and jail offenders for this, or Congress permanently loses the power of subpoena moving forward. As bad as the former may be, pretty much everyone would agree the latter is worse.
The DOJ isn't the sole source. They can turn to any law enforcement agency, theoretically. The obvious one would be Washington D.C.'s police force. The District is technically under Congress' purview, so it's not even a stretch; the Washington D.C. police are more properly the enforcement arm of Congress in this than the FBI would be, really.
And D.C. already has jails, so that's covered.