Well, I do not use Live Track, but the phone app continues receiving live data (i.e. HR, steps) from my watch without problems when BT headphones are connected. Is it the same case with you, or nothing is being transferred at all, when using the headphones? And when you go to BT details in the phone settings, does Garmin remain paired, when the headphones are connected?
You could try clearing the BT cache, to see wheather it helps. You can also try googling something like "phone bluetooth multiple devices conflict", better yet adding the name of your phone, to get more specific hints. You can find many tips for troubleshooting this kind of issues. You could also try it with another model of headphones, to see whether it is a more general problem, or rather a more specific conflict.
Some tips are for example here:
https://www.techlicious.com/how-to/how-to-fix-bluetooth-pairing-problems/comments-/CP9/
https://thewirecutter.com/blog/understanding-bluetooth-pairing-problems/