I've taken a similar approach with the BOSS RC-3, but I upgraded to a BOSS RC-10R, which is still small enough and midi compatible. I use it with my FM3 currently, but no doubt it will definitely work with the AXE III.
I just through a midi block in the preset and control the presets, start and stop via midi block with the scenes. Mostly I set the sample in the RC-10R to one-shot but could use the cc in the midi block to stop the sample/loop.
The BOSS RC-5 would work well too, it's even smaller with midi capability. I'm kinda wishing I went with that option instead. But if you want the beats and rhythm section too, the RC-10R is the way to go.
I think that's your best bet for an immediate workable midi solution. I really wouldn’t see Fractal adding anything like this anytime soon. I imagine they would need to add an option for external memory storage to pull this off. But I'll leave that to the armchair steak-head engineers to argue about.
You'll need to set up the BOSS unit with which cc you'd like to use for "start" and "stop", so it's really up to you. I chose 6 and 7.
Here's an example of a scene selecting preset 2 of the BOSS unit (midi offset is by 1) and the backing track start is set to cc 6.
View attachment 81917
As mentioned, I normally let the backing track play out as a single-shot, but here you can see the next scene that stops the backing track if that's your desire, the cc is set to 7. It could even be thought of as a panic kill switch, which is nice to have for false starts.
View attachment 81918
I too waited a while for a good viable and simple compact midi solution, so I hope this helps you. Also, a lot more can be done with the midi options, but this covers my simplistic needs.
Good luck!