This is a poor solution to a non-problem. 99% of "private networks" don't need HTTPS for most things, full-stop. Those that do, the correct solution is to deploy a private CA and use internal DNS. The implication being you do not trust your network. So the solution is to.....use public DNS and Let's Encrypt garbageware and leak details of your internal network so you can pretend you're now more secure because the cafeteria menu is hosted over HTTPS? Save the black hats some time and just email them the Visio of your core network.
If you don't have access to install the root certificate then by definition you don't control the private network, simple as that.
> when you want to share the application with others
This is literally the definition of no longer a "private network".