The plywood did the trick.
I went down to the lumber yard (1/2 mile walk), picked out a sheet of plywood, got them to trim it to the right dimensions for me and deliver it that afternoon. Whacked it on the bed with the help of the Pirate when he got home from swashbuckling, and lo! A (mostly) firm bed!
It's still a bit spongy, but eliminating the give from the box spring has done a lot to firm it up. The big change is that now I can turn over at night, which I couldn't before. That helps a LOT.
So this will get us through at least until the New Year. Pirate has finally been persuaded that we need a new bed anyway, if only to upgrade in size. After I spent three nights on the floor* and he had room to stretch out, I climbed back in and he declared the bed too small.**
*In my nest. I slept on my camping bedroll between the bed and the wall, a space about 2 feet wide and 8 feet long. I had 3 pillows (two for under my head and one for hugging***) and burried myself under 2 down duvets (one of which was king size and folded in half). It was a proper little nest. I quite enjoyed it.
** Which I told him when we registered for the duvet on the wedding list. I insisted on getting a king size because I knew we'd need a king-sized**** bed as soon as we could afford it. He scoffed. Ha!
*** this is critical
**** A king size bed in the UK is equivalent to a queen size in the USA*****
***** I'm trying to beat GSE for the Most Footnotes In The Footnotes Prize