The user should be in control of where the file is placed and putting it in /tmp by default is probably not the best idea anyway because if you download and open a score, you probably want to keep it and /tmp isn't meant for persistent storage. If the way musescore works is if you want to keep it then you must 'Save As' (or something), then that should still work fine-- you save it to home somewhere.
Why is firefox defaulting to putting it there? Firefox has a way to configure this (about:preferences, then under the General tab). Two things seem to influence this: setting the directory and setting to always ask where to put the file (see https://askubuntu.com/questions/816983/firefox-browser-ubuntu-converting-all-my-downloads-to-tmp-files). Perhaps the musescore site is written in a way that it makes the files look like temporary files instead of regular downloads (I don't the site or how to influence firefox in this regard...).
As for the general situation of sharing files via /tmp-- this is not supported at this time for the reasons I mentioned before because /tmp is something different to the snap and to the system's firefox. It is technically possible to make using the per-snap /tmp optional, but I'm not sure this fits into the project's plans. At a minimum, it would be confusing that some apps could see the system /tmp and others could not. An architect would have to weigh in on this.