Since their modem is handing out DHCP addresses, is there any reason why you couldn’t just connect that cable to your router’s internet port, and configure it for DHCP on that interface? Then the provider would always see their modem, and you’d still have functional routing that you control.
Since consumer routers have a dedicated interface for this, you don’t have to make routing tables to tell it which way to the internet, it already knows it’s all out that interface.
Just make sure your router uses a different private address range for your network than the one handed out by the modem.
So your router should get a DHCP and DNS settings from the modem, and will know it’s the first hop to the internet.
I do this to create test networks at home (my cable modem has multiple ethernet ports), using cheap consumer wifi routers. By using the internet port to connect, I can do some minimal isolation just by using different address ranges, not configuring DNS on those boxes, and disabling DNS on my router.