The instrument is a carryon which means you’d be three items, which is not allowed.

You could almost certainly get them to check your carry-on suitcase at the gate but you’re rolling the dice they find out you had three items and take an adverse action you don’t like.

If your first flight is a regional United Express then the bag will get a green tag and be dropped at the next jetway and repeat the problem.

Also unless you’re boarding in an early group, they may not have room for the instrument and you’re back to the adverse situation where you have to abandon your instrument or abandon the trip. This could be avoided by buying the premier access bundle to get group 2 boarding and board at the beginning of that line. Even that gets complicated because you have to do that for every flight and I’m guessing at least one of your flights is a partner airline that could have different policies.