That's just whacked that cable companies are now using combo Modem/Routers. What a PITA. My FIOS hookup also requires the companies modem/router. Actually, I'm not even sure it qualifies as a modem, as the ONT can have Ethernet out, but since I have a TV package, the ONT has to be configured with coax out, and then their router converts that to Ethernet. I have my own router connected to that router.
not to sidetrack
You can most definitely use your own router instead of the FIOS router, you just need to be converted to Ethernet from the ONT. I had FIOS since it launched and even after getting the TV service a year later when it became available i was still on Ethernet to the ONT. Check out the FIOS dslreports (eh broadbandreports now i guess) forum on what to ask the CSRs to get switched over.
I used a smoothwall express 3 setup that could dial pppoe (what they used with initial rollouts, its DHCP now i believe). I then put there router on my orange (internet access but no internal network access) LAN for VOD and other network data the STB's used. My only drawback was not being able to use the FIOS smartphone remote app since I was on a different internal network (orange vs green).
I do egress firewall filtering and didn't want to deal with maintaining a firewall ruleset for the their STB network calls. Also didn't want to try and figure out all the ports and protocols needed to fix any features (VOD, menu data, etc) I may block with my firewall.
on topic:
If you trust the new device as the new NAT'ing firewall bridge them as switches not as routers if your router has an AP (access point) mode. Connect the LAN port on your linksys to the lan port on the new router/modem. Make sure only one of the devices is providing dhcp, and make sure you linksys LAN network matches what is now provided by the new device.(eg 192.168.0.x vs 192.168.1.x).
This might work even without an access point mode.