I brought this up in IRC today (and I think I've also brought it up before), and was recommended to bring it up here. I'll try to summarise this.
I'm aware that there's been some friction between the Palemoon development team and the maintainers over at F-Droid regarding their Official repository. As a result, Palemoon is not available therein, and this situation is not going to be changing any time soon.
However, there is still interest among many Android users (of which it seems I'm claiming to be somehow representative) to be able to install Palemoon from a source that can provide official, automatic updates (so, not just installing the raw .apk or self-building), but without having to marry themselves to the Google infrastructure.
It seems from the discussion in IRC that the Palemoon maintainers aren't aware of another option - it is indeed possible to have Palemoon installable through the F-Droid package-management software by non-GApps users, but without Palemoon having to have any ties with either the F-Droid maintainers or with their official server: it is possible to set up your own F-Droid repository: https://f-droid.org/wiki/page/Setup_an_FDroid_App_Repo
After you set up such a repository (which from their guide, seems quite simple), all a prospective user would need to do is click your "F-Droid" link on your page, have your repo with its key added to their list of package-sources, and then simply install Palemoon from it using the F-Droid software.If you host your own F-Droid repo, then people can use F-Droid to install your own builds signed by your own signing key
So... could this be an option?