We encourage experimentation here so if you build this way, and it works then document it so it can be shared.
I do see some possible issues that I will share with you.
If I understand correctly, you want to make an rmh
and plumb it into a half barrel system. All good.
Next you are thinking you want to bury that 1/2 barrel 3-4 ' deep. Sitting it on top of perlite for insulation.
This is a problem area. To bury that mass in the ground, it must be heavily insulated (think
4" foam board) on the bottom and up the sides to ground level.
Any less and your heat will absorb into the earth (Nothing but the sun can heat the earth) .
If you insulate your mass from the earth and then pour your slab over top I think you will not be very happy with the result either.
To start that slab would need to be every bit as insulated as the mass was. If you do that, I suspect that you might see a temp climb on the slab but it will be minimal and not the toasty warm pad you are hoping for.
Leaving your insulated buried mass open to the room (not under a slab) would release more heat into your shop. But mainly your heat will be coming from the barrel over your riser.
The buried mass will radiate but only from the top not the sides.
If it were me I would build an above ground rmh with a brick bell, you will be finished quicker and your room will feel warmer.