More cycles of 'spray, sand, sigh' have happened. In the end I used 3 cans of primer, and had 4 attempts at spraying and sanding before calling it 'good enough'.
The 3rd attempt was a bit of a nightmare - the 3rd can started out spraying really dry stringy stuff, it was almost like silly string coming out of the can. The primer ended up looking really gritty and bumpy. Sanding it back, it had air bubbles all through the primer, and in the end we sanded it all back to the 2nd coat. It ended up OK, but there were a few big areas on the front and back where we sanded through to the wood again.
I was going to leave it at that, but a friend who makes movie props convinced me to try another coat of primer. The 4th attempt went much better. It was the same can, but seemed to come out really nicely. Not sure why. Maybe I shook it up better this time.
After another 1.5 hours sanding, the primer coat was done. A few tiny bits of overenthusiastic sanding on some bits of the edges, but for the most part it is good.