Amazing project! Thanks for reaching out, I would like to help where I can. I agree, it should be 100% doable. HTTPS will be the tricky part. Is this mesh work not connected to the internet at all? It would work with plain HTTP, but there are risks for the users that you would have to weigh.
You can get started very quickly, just install Debian/Ubuntu etc. on any computer, install any webserver (Apache, nginx, lighttpd, etc), then use fdroidserver to index and publish the APKs to that webserver. It is also possible to use an Android device as the webserver, the fdroid update publish process would still need to happen on a computer somewhere. For that, there are webserver apps for Android.
I can help you with downloading APKs and updates as needed. I could even mail you an SD Card with a whole 64GB copy of the f-droid.org repo.
In any case, we welcome your questions.