For people who are interested, I will share the 16K internal upgrade with a 128K SRAM. I didn't have a 62256 SRAM in my parts bin, but I did have a 128K SRAM. I decided to perform the upgrade with that.
I compared both datasheets and it turns out the pinouts are almost the same, the 62128 has 32 pins, the 62256 has 28 pins. That requires a little modification to make them work in a ZX81.
The modification is quite simple:
Cut pins 1, 2, 32 and 31 so they cannot short out on the PCB.
Take some wire and solder pins 1 & 2 together, wrap the wire around and solder pins 32 & 31 together and connect them with pin 30. This will tie them to +5 volt, effectively pulling them high.
Bend up pins 3, 28, 25 & 23 (1, 26, 23 & 21 on a 62256)
Solder a wire from pin 3 to the cathode of D1
Solder a wire from pin 28 to the cathode of D5
Solder a wire from pin 25 to the cathode of D3
Solder a wire from pin 23 to the cathode of D2
This will give you 16K of internal RAM on your ZX81, I found the wiring directions here:
http://blog.tynemouthsoftware.co.uk/2017/10/zx81-internal-16k-ram-reversible-version.html
I recommend soldering a 10K Ohm resistor between pins 29 and 30 (27 and 28 on a 62256) to make high resolution stuff work.
That's it basically. More RAM is possible, but that requires additional logic.

The second ZX81 which I got for free, this an Issue 1. It had a broken ULA, which was a 2C184E. Replaced with the awesome VLA81. This little machine now has 32 KB internal RAM. All of the needed logic and decoding is done on the VLA81.
Also replaced the voltage regulator for a switching type and replaced the electrolytic capacitors.
If we had known back in the day what was possible now.....