Everything is controlled by how the song is set up in the CK. You could even set up a track to only play for a few moments just as you're looking for.
To make the music stop you can also have it play a blank track (A track of recorded silence). It takes some messing around with the tracks to understand how they work.
As far as "push-to-game" that seems to be more of a Skyrim Special Edition error. Did some research for you (not much) and found this same problem with SSE over and over.
There was also no solution to any of the posts (90% didn't even get a reply) and they mostly were problems with Cue Points.
Take a look at my Enchanted ship in a bottle mod. In the downloads I have a download for all the mod scripts (a wealth of mod scripts for home mods). In there I have a duel music player.
It will show you how I was able to switch tracks, fade them in and out and keep track - of 20 music tracks. Took me a while to figure out a new way to do this as I too was having problems
with the cue. Also download the ship mod, load it up in the CK and take a good look at how the tracks were set up. This was the key to everything. If they are not set up that way they will not cue correctly.