I'm trying to get the compile prompt to suggest completion based on past
commands I've run.

You can use 'C-x <up>' in the compile prompt to complete on past commands.

This is indeed a lot closer to what I want, however it still lacks a lot of things compared to my solution.

Unless there is some package/mode I don't know about, It's a lot less dynamic, for instance I can't keep typing to reduce
possible items.
My solution can leverage all of vertico and friend, and in general provides an experience that's the same as other minibuffer prompts, like M-x.