Is the cable modem performing NAT translation for the 192.168.201.0/24 network, or merely routing traffic which it won't do on it's WAN interface since it's in a reserved private only address. In other words, does it only perform NAT translation for the network its LAN switch is on?
If that is the case, maybe treating the 192.168.201.0/24 network as a the DMZ could trick the modem. to provide NAT if it allows a DMZ port configured on a separate private network.
You could configure the Linux router to masquarade addresses when the source is from a 192.168.201.0/24 address and the destination isn't a local address. So you would only be doing, what you want to avoid doing, if the destination is on the Internet. You could NAT to a single address. This would be using double-nattng for the hosts on the 192.168.201.0/24 network.
Another option could be to subnet the 172.20.101 network with the host addresses changed from the 192.168.220 network to the top subnet. The router's LAN interface would still have /24 scope encompassing both /25 subnets.