You could use 1/2 extruded polystyrene foam board Insulation as the inner panels of the room.
They are only R3, I think, but they are advertised as impervious to
water.
That's about $182 for 13 sheets at Menards.
The 13 sheet is to create strips that go on the face of the studs.
That will cut into thermal bridging.
You could use adhesive and skip the screws, since the insulation board is light .
If you do skip the screws, be sure to also use free standing shelves that don't need to be screwed to the walls.
You could use R23 Rockwool in-between the studs.
Going off Menards prices that
should be $800 for 16 packages.
Rockwool can get wet ,then dry out and still be insulative.
I would probably use the sheets but fill the wall cavity with Styrofoam waste.
For cladding you could cut up an old fridge and screw the panels to your studs, sealing the cracks with expanding foam.
That will look terrible but it should work pretty well.
You could actually build your entire structure from old refrigerators, including the foundation .
But that's a lot of bootleg for a
project that you need to work.