Did this on my static van's utility room extension a couple of years back — timber frame, basically the same principle.
Went with Rockwool RWA45 between the studs, then a layer of Celotex GA4000 on the inside face before boarding. The combination of mineral wool (for acoustic/fire) and rigid PIR meant I could keep the stud depth reasonable without losing too much internal space. Total cost was manageable because I bought the Celotex as offcuts from a local insulation merchant — worth ringing round, they often have job-lot remnants going cheap.
Key things I'd flag:
- Vapour control layer — don't skip it, especially if you're running the place intermittently and letting it cool right down. Condensation in the frame is a nightmare to deal with later
- Thermal bridging at the noggins is where a lot of builds lose performance. That continuous layer of rigid board on the warm side really does earn its keep
- Airtightness tape on all the joints, even if it feels excessive. Drafts kill efficiency more than thin insulation does
I priced spray foam too — the closed-cell stuff is brilliant for awkward shapes — but the quotes were eye-watering for a small area, so batts and boards won out.
What spec are you building to? Are you going for anything close to building regs U-values or is it more a "warm enough on a budget" job? Makes a difference to what's worth spending on vs where you can cut corners.