Certificates for hosts on private networks

Actually could Let’s Encrypt help with the second path I mentioned above.

That second path is as follows

My native app (OSX/Win/Linux) is also used in a bunch of museums and installations. In that case there is no internet. Users connect to the WiFi, using fake DNS and Captive Portal support they either wait for the captive portal stuff to kick in and get automatically connected to the installation (iOS) OR they type any domain name like h.com and get redirected to the installation (android/winphone).

As there’s no connection to the internet I’m curious if there is a possible solution.

Basically instead of me having to provide a certificate museums can get a free one from Let’s Encrypt and install it locally in the app. Maybe they can register it to a domain like “someinstallation001.museumname.myapp.com”. I can then make the app (which is faking DNS) redirect all users to someinstallation001.museumname.myapp.com which is still just the same server but as it would be providing the correct certificate would that work completely disconncted from the internet?

Would that work?