Thanks for your reply, Howard.
I think I probably could drill the pin out with a bit of effort. The more difficult bit might be replacing it with a ball-bearing that functions as I want it to. I suspect choosing the size of the ball is critical: too big and I risk damaging the hole as I drive it in; too small and I may not be able to move enough material in from the lip to hold the ball in place, and once I've started to move material it may be difficult to go back. The good news is the ratchet as it is functions quite well with most of my 3/4 sockets (i.e. it still holds most of the sockets on the drive, albeit a little tenuously).
Incidentally, one of the two countersunk slotted screws you can see holding the cover plate in place is an impostor. One of the screws was missing when I got the ratchet out. I rifled through my boxes of misc. machine screws to try and find a replacement, which needed to be a 24tpi 3/16. The only 24tpi 3/16 screws I found had hex heads on them and were 1cm too short for the job.
The solution (I don't have a lathe): cut the hex head off the screw that is too short; cut the head and a bit more off a slotted screw of the same diameter but a different pitch, and bronze-braze this head and a bit more on to the decapitated replacement.
It works fine, though having a lathe would have been nice. I mean to fix my lack of one soon.