The FIRST thing to do {if they want re-election} would seem to be slash the Travel and Junketing budgets of ALL elected Federal Office holders.
ONLY “tourist class flights”; until a budget is in place.
Likewise, Congress can cut the money being spent on the Executive Office of the President (staff). By cutting the funds Congress could shut down or squeeze departments and agencies like EPA, Education,and even Obamacare itself.
Congress has the power to inflict great pain and embarrassment on a president that will not play ball. They can charge the president with dereliction of duty for not submitting a budget as required by law. The House can bring charges against the president. The Senate would have nothing to say (no vote) on bringing charges.