That set-up using a different battery could work for me. I see this gizmo, but it is not clear if its recharger could keep up with the draw of the phone company's equipment via its inverter so I could just leave it plugged in same as my jury rig.
I don't know if the modem would get knocked out in "one cycle" or not. It takes a minute or two to get the internet back once it has power. Leaving it on the inverter all the time avoids that.
https://www.canadiantire.ca/en/pdp/motomaster-eliminator-powerbox-600-0112002p.html