I bought a 16K SRAM pack from https://www.youmakerobots.com/sinclair/85-zx81-16k-ram-kit.html on eBay for £18.

It uses this circuit:

I assembled this and it worked fine.

The board can be modified to provide the full 32K of the RAM chip by driving RAM pin A14 from /M1 * A15.
This forces RAM A14 low when "running" the display.
The ram at $C000 to $FFFF is now not an alias of the user RAM but an extra 16K RAM that can be read and written to, but machine-code cannot be run there.
The right-hand pair of NAND gates in the LS00 chip have floating pins, which is a bit naughty but you can get away with it with TTL chips. Having floating pins makes the 32K modification easy, because the pins can be simply wired up.
The RAM A14 pin is grounded, but thankfully has thermal breaks so you only need to cut four small PCB tracks.

A further mod would allow you to relocate the ZX81's on-board RAM. Maybe not worth it for just 1K (although the G007 board uses it). However, if you have up to 8K you could re-map that at $1000-$1FFFF. Or re-map bigger memories to $8000-$BFFF as well. such mods would require more logic than this 16K RAM board has available.