I recently had a similiar problem, posting the solution here in case it helps others.
My inky impression 13.3" worked fine when connected directly to a Pi Zero 2 W. However, when I powered the Pi from a Witty Pi 4 mini (I rewrote the firmware to avoid the pin conflicts), the image would appear to transfer to the screen OK, but the screen would become unresponsive.
(The unresponsiveness was masked by Busy wait bug on Inky 13.3 panel · Issue #254 · pimoroni/inky · GitHub however, but I patched this out in my own code).
After such a failed transfer, the screen would enter a state where it would remain unresponsive, until you left it for a while (maybe 30 mins?), when it would recover.
I think what’s happening is that the screen draws a lot of current from the 3V3 rail when it starts to refresh. This rail is supplied by the Pi, and if it can’t keep up, one of the control chips on the screen browns out. After this happens, it takes a while to recover (presumably because there’s some capacitance on the screen PCB which takes a while to drain). If the Pi has a good 5V supply it’s fine, but if the supply is a bit limited (such as from a Witty Pi 4), you hit problems.
Sticking a 1000 uF cap between the 3V3 and GND connections on the screen (I used the header in the top right) seems to have fixed the problem!