Verizon routers creates a separate, hidden WiFI network for FiOS ONE boxes. They only work on this network. I don't think any non-Verizon mesh system will extend this network. Certainly a non-Verizon router won't even create it.
Verizon does sell a network extender. This may extend the hidden network as well, but I have no direct knowledge if it does or not. Hopefully someone else who has the extender and FiOS ONE boxes can comment. Here's the device:
Know that using a non-Verizon router with FiOS TV can be complicated. The FiOS ONE boxes are even more dependent on a FiOS router then the older boxes. There are reports of people getting it working, but they have to put a Verizon router back in the network for activation and software upgrades. Unless you like periodically troubleshooting and tweaking your network, I suggest you stay with a Verizon router.
The other option is to use a wired connection. The FiOS ONE boxes can work with coax or Ethernet. If you have either of those available, I suggest you use it. Wire connections are always going to be more reliable then WiFi.
The G1100 should be able to handle the load you have. It's WiFi range isn't the best, though. The G3100 is reported to have better WiFI (it certainly has the latest, fastest WiFi specs), but I doubt upgrading to one is the only way to solve your issue. I have not personally tested a G3100, though.