The democratic party is actually actively promoting the "President Musk" language for that exact reason. They're also trying to aim to say it during appearances where Trump is watching
The “President Musk” messaging is by design, at least partially. This week, according to a source with direct knowledge of the matter and another person briefed on it, at senior levels of different Democratic congressional offices, and also within the Democratic National Committee, discussions have been had about having party leaders and elected officials actively portray Musk as effectively Trump’s boss, and to do so during television appearances that the president-elect is likely to see. The idea is that it’s a cost-free opportunity to potentially drive a petty wedge between the notably mercurial and ego-obsessed Trump and his similarly emotive pal Musk, and to sow some chaos in the upper ranks of the Republican Party.
It didn't include the debt ceiling raising that Trump & Musk blew things up for. They were able to get some of the funding changed in the second version back. For instance, Senate dems were able to do some maneuvering to get back most of that pediatric cancer funding that president-elect Musk and house republicans cut of their CR bill
The house had already technically passed an earlier bill that approved similar funding (intended to be merge into the main CR), so the senate dems just went back to that earlier bill and voted on it and got that passed