Yes exactly, please try disconnecting the display from the Pi and test the Pi alone. Maybe there is a problem with the power supply when the display is on. Please do not use Multisqueeze to test streaming to the server via Bluetooth. Please just use the Pi, the jack output of the Pi and Squeezelite. If it doesn’t work (which it definitely should, after all I tested it again on site with test devices), then there is some kind of misunderstanding. I don’t think it’s a hardware problem (except for the display maybe).
Bluetooth should not affect the headphone output, of course. Please make sure that in the advanced settings of Squeezelite „sysdefault…“ (without „…hifiberrydac…“) is selected to use the jack output or HDMI for audio output. To be on the safe side, you could also burn a fresh image (Buster) and follow the 10 steps I sent you the other day.
I know it’s frustrating when something like this doesn’t work even though it should. But I can’t do more than give you these tips to solve the problem yourself.