So I've had this issue on and off for years, across 2 iPads, but just worked around it by opening the app directly and copy/pasting manually.
I spoke to support 2022-06-11 and they advised it's some kind of issue with the external keyboard. I was pretty sure it happened both with and without the magic keyboard attached in the past, but today on the newest (5.12.0) version of Lastpass I actually found that auto-fill stays open as long as I need it to, as long as I'm not attached to the external keyboard.
TLDR: Seems to be "fixed" by temporarily removing the external keyboard.