I'm thinking the units communicate via bluetooth. Correct? Since you said BT is fairly useless......no big loss.

Well, you might have to look at using some sort of splitter such as an external mixer of small size. Input the mic, guitar, and backing source into it. Send the output via a hard wired connection to the amps..... split with a Y-cord or even something like a 4 channel headphone amp. I use a Behringer 4 channel headphone amp here in my studio. One input gets me 4 outputs that I can control the volume of each output uniquely. $50 for the headphone amp.

To move forward, you might have to take a step back to old school technology and use wires.

Set up for the larger jobs might take a few more minutes, but the smaller gigs remain the same.

You can find my music at:
www.herbhartley.com
Add nothing that adds nothing to the music.
You can make excuses or you can make progress but not both.

The magic you are looking for is in the work you are avoiding.