One fast, easy way to ensure prompt release of the Mueller report would be for the House Judiciary Committee to start explicitly considering articles of impeachment.
In calling on Representative Adam Schiff (D-CA) to resign as Chair of the House Intelligence Committee, Representative Mike Conaway (R-TX) asserted that the report “conclusively refute[s]” allegations of collusion between the Trump campaign and Russian agents in 2016 — this despite the fact that he has not read that report any more than anyone else has.
If the report is as clearly exculpatory as Republicans claim it is, then they will certainly want to offer it to the House Judiciary Committee to stop consideration of impeachment. They would have no better ammunition at their disposal.
Call their bluff. Impeach the president.