If the team complained about their code base for _years_, I'd say the quality of the code base they started out with wasn't the only problem. Especially considering that you had the resources to write an email app from scratch.
While it's true that parts of K-9 Mail's code base are gruesome (even today), I can't remember any significant contributions from teams that have built products based on K-9 Mail.
Switching between the F-Droid version and the official APK (Google Play version) is not possible. You'll have to uninstall the old version first (and that will remove all settings). However, K-9 Mail supports settings import/export.
I understand it's difficult for a company to give money to an open source project. But none of the options you've mentioned sound very appealing to me.
GitHub displays the avatars of sponsors. Maybe that's an option that satisfies the "acknowledges sponsorship" criterion?
https://github.com/sponsors/cketti
> I definitely struggle to understand why IMAP IDLE is critical for a release.
The functionality is present in the latest stable version, but needs to be reimplemented in the new version. As a user I would be upset if the app was updated and that feature taken away, even if it was only temporary.