I think that one of 4464 memory chips is failed. Find it out and replace with working one (any 4 x 65536 bit chip should work)
Also you can try to swap two chips on card and if you notice some changes in picture you have probably found a defective module.

It looks like the 4464 is a one-bit memory chip

No, it is 4 bits x 65536 memory chip, so 8 chips will result in 256 kb of VGA memory with 32 bit bus width.

It looks like the chips that hold the highest four bits of each 16-bit word (so two chips for a 32-bit bus or a single one if the video controller runs on a 16-bit memory bus with two banks). It might be better if you desolder the chip and add sockets to the board.
Also, I'd try diagnosing it on a high-res graphical mode that makes use of the full memory range of the card. If only half of the screen gets corruption on every odd pixel then an educated guess would be that there are two memory banks and only a single chip is busted, and swapping the banks would allow you to use VGA modes just fine (unchained mode apps can still face corruption)

Thanks for the suggestions! The chips are soldered on, so swapping them isn't straightforward unfortunately. Replacing them with sockets might be the way to go.

Since I've been investigating, the characters have now become corrupted as well, and the system no longer boots - no memory check, so best guess is the VGA BIOS is displaying an error message and halting (it looks like a box has been drawn on the screen but I can't make out what letters are inside the box.

If I'm going to have to desolder all the chips, would it be possible to replace them all with a single modern memory chip? Or maybe two if there are two banks? I'm guessing I can just hook up the address and data lines to match, and providing the replacement chip is 5V tolerant and fast enough, it should work?

You might be onto something with the two banks - the chips are labelled U9 U10 U11 U12 U1 U2 U3 U4 in that order, and it looks like there's a break in the address lines between the two groups of four chips, so it may well be in two banks of 16-bits (which would again make sense if only the attribute bytes were originally affected, as they appear in the same position every 16 bits.)

If you find a modern 5V chip that has simillar characteristics, then it might just work. If you have a programmer like a TL866, I think it can also work as a DRAM tester for 4464 ICs - I used my TL866 to isolate a faulty cache chip a while ago. If you do desolder and put sockets, try swapping the banks - maybe that will be enough to get it to boot.