I found somewhere to order a filler primer from and have a couple of cans on order.
Looking at the state of the plastic even after sanding, it's gonna annoy me, so hopefully this helps take the edge out.

Progress update:
All of the patterned outer components are now down.
Waiting on:
- Centre - currently printing. At 100% infill, this is about 390grams of plastic alone.
- Centre neck. That's next. It's bigger and heavier, and I'll need to change filament before I do this bit.
- Glue. It's coming. The original build guide suggests gluing from the centre out, so I'll go with it that way.
- Neck. I may have ordered a black neck that's in transit. Oops.
- Headstock logo. I'm gonna try something different and have a raised logo and axe name on the headstock. I'll 3D print them in white, lay down a coat of polyurethane and see if that'll hold it. Then I'll see what happens if I add a few extra coats over the top. A little experimentation involved, but it should be super light so I imagine if I put it down as soon as I've put on a slightly heavy coat, it might be enough to hold.