While it would be lulz worthy, it'd also throw an brick into any 'real' legislative process for a while, and I'd really rather not interrupt my enemy while they are making fools of themselves, yknow. An impeachment move on weak ground would just give the media an excuse to suddenly ignore all their current failings, would give the RINO's a chance to curry favor to reinforce their position by going against it, and would give the Dems an excuse on why they haven't done shit to help people for however long the process takes, even if they could totally focus on other things while it happens.
Better to leave the narrative focus on the moratorium failure, the disaster that its creating, and the fact that they are squarely holding the bag. No need to give them a convenient smoke screen.