Alesis Multimix is a small USB enabled mixer with one of the input compatible with "instrument out". So basically, it is the same thing as for example M-Audio M-Track-Duo, just costs twice the money. With that you'd still want some kind of amp simulator which the Joyo Brown is the cheapest while an ok thing I can think of. Well there is also a clone of it called Caline Wine Cellar for $10 cheaper. Without this box, the tone of the bass most likely will be crap, but you can try without it and see how you like it. The cheapest setup I can think of is probably M-Track-MONO and Caline Wine Cellar. That will set you back probably $90 from Amazon (or just the M-Audio piece for $50).
The main problem that you have is the need in connecting your headset in. If it has a cable, you can connect it to the headphone out on the M-Audio and hear stuff, but microphone is a problem. Even if it is available on its own connector from the headset, it is unlikely to work connected to the mic input on the M-Audio as it needs special power which neither M-Audio nor Alesis will provide. So I do not see any way not to have a microphone that can be connected to the XLR input on the M-Audio. Only other way is that you switch the "microphone" device in the Zoom by hand. You pick USB interface for playing, and when you want to talk, you pick the bluetooth. Probably not ideal.
The cheapest mic that you can plug into M-Audio will probably be $30 Amazon Basics Dynamic Microphone which comes with an XLR to 1/4" cable which you can plug into the M-Audio Microphone. So at a bare minimum you will need some kind of USB Audio interface with an Instrument input, and even the crappiest one from AliExpress will be $30 range. And the $30 mic... And something like Caline Wine Cellar or Joyo Brown (Sparrow) to improve the sound of your bass.
You can also just mic the Amp/Cab. Get something like Samson Meteor USB mic, put in front of the cab. When you talk to your instructor over zoom, you use your usual Bluetooth headset. When you play, you select the Samson as mic device for your zoom. Actually if you set it in omnidirectional mode it probably will pickup enough of your voice as well. Not optimal, but will work better than your current headset alone.