I would think that any wireless router would work.
If there is a catch, it will probably be in the setup of the router to make it think that the internet connection is just "temporarily" missing.

and any decent wireless router will let you do the home network, no internet needed.

Your devices should be able to talk to each other.

Netgear N600 is a place to start. There are many similar ones around.

L-

Any wireless router will route local traffic with or without a link to the internet. And yes, DNS host name lookups will fail, but you wouldn't use those for local machines in any event.

Mac's (and iOS) have "Bonjour" to find local devices (independently of DNS), and I'm pretty sure you can get it on Windows & Android if you want it. You don't really have to have a local DNS server, but it might help keep things sane. Worst case is you get to configure a fixed IP address on the thermostat and printer and tell your other machines what IP address to use for them.

Be sure and set up both 2-gigabit and 5-gigabit wifi. The 5 is good for fast & nearby. But 2 works better for longer distances. You can pick the same WiFi name for both, and your devices "should" automatically switch to the stronger signal as you move around the house.

Btw geosync satellites can be used for internet, even way out in the boonies. They used to be really awful, but the recent ads I've been getting make them look a little more reasonable. Of course, not good for gaming. The latency to geo-sync and back is really slow. But for web browsing and occasional downloads, you might find them useful.

The N600 mentioned above does. Just need to make sure you turn you phone hotspot on when you want to use that wifi for other devices. (Bridges and extended via the router)