With not expensive non USB Mixers you can switch the return inputs to the mixer inputs or the return inputs to monitor.
So if the returns are selected to the mixer input - it goes back out with the mix - not what you want.
If you select return to monitor - then you can't hear the inputs - again not what you want.
From my very short look at some usb mixers - its the same.
You would need an expensive mixer with multi channel out and internal multi buses. (JimN comment)
Taking the lead from Artyman - I plugged in my Behringer Guitar USB interface.
This is then similar to plugging in a USB Mixer.
The latency with Audacity is very high. So its impossible to use the return signal from Audacity anyway.
So the only way to do it is to run the BT from Audacity into monitor speakers of some sort - and use your guitar amp speaker for you - with a mic or a direct input.
Your accompaniment is recorded - with latency which is corrected by audacity when the number is completed.
The amount of correction has to be set up. Artyman does this. I think you record something and use the waveform display to find the latency and put that in preferences. (Someone please correct me if I am wrong).
If you have "Playthrough" selected then you will hear yourself - about a week after you hit the strings
(Latency)
General note:- I use the Guitar Interface for live playing with Cakewalk DAW. Latency is 9ms - so the latency is as I understand it is down to Audacity.
I know many people have had very good results with Audacity - and its free.
I hope this helps.