When Tesla installed my Powerwall, they suggested that I don't use Wi-Fi, even though I had a mesh node in the garage near the Powerwall. Instead, they used a set of TP-Link Ethernet powerline adapters to connect to a RJ-45 jack directly on my router. I have never had any issues. Could this be an option for you?
https://www.amazon.com/TP-Link-AV600-Powerline-Ethernet-Adapter/dp/B00AWRUICG
I got the Cyberpower ones from Costco for around $125 for the sine wave support (to protect electronics).
If you want one for a fridge I would suspect it wouldn’t last very long, though if your only true concern is knowing when power goes out, they do sell little alarms you can plug in for that. See below:
https://www.amazon.com/Powerout-Power-Failure-Alarm-Safety/dp/B00PUQILCS
If you really want to know, look into something like this power monitoring system.
Some good suggestions here already but how about a MiFi or similar device? Essentially a cell powered wifi hotspot. Something like a Verizon MiFi 6620L would probably do the trick. You would just have solid power to it and make sure its secure. If my Unifi stats are accurate my single Powerwall only uses about 3gb/month in data which is something a 15gb/month SIM would be more than plenty for (Verizon's "essential" plan is $20/mo for 15gb/mo).
Obviously a bit more expensive to set up and maintain than the other options but would be 100% yours and completely separate from your renters. If that is valuable to you it would seem the price may be worth it.
That is a challenge, as Tesla keeps breaking the auth code for many apps. The easiest I know to recommend is the Tesla Tokens app for Android; I believe something similar exists for iOS.