I know that in the US there’s a law requiring that airlines let instruments into the cabin if there’s room when you board the plane:
“Carriers must allow passengers to stow their small musical instruments in an approved stowage area in the cabin if at the time the passenger boards the aircraft such stowage space is available. Under the rule, musical instruments as carry-on items are treated no differently from other carry-on items and the stowage space should be made available for all carry-on items on a “first come, first served” basis. Carriers are not required to give musical instruments priority over other carry-on baggage, therefore passengers traveling with musical instruments may want to buy the pre-boarding option offered by many carriers to ensure that space will be available for them to safely stow their instruments in the cabin.” (from transportation.gov)
However, there’s a chance that if you aren’t permitted a full-size carry-on, the airline has assigned you to the last boarding group and it’s very possible there won’t be room in the overhead. In any case, I would contact the airline beforehand and ask if they will allow you to bring on a musical instrument in addition to your personal item. I would feel free to mention the law when you ask.
The size limitation on a “personal item” vs “full-size carry-on” is generally shrinking because airlines want to charge us more, but they’ve always let me on (so far) with a modest backpack. Most airline limits seem to be around 18"x14"x8" here in the states, at least for now.
I hope you find an excellent solution! If I knew more about specific cases I’d definitely share. Good luck!