Can I subnet a subnet?

That all looks perfectly correct. Note that the servers will use the .240 netmask and either .164 or .180 as the gateway. However, are you sure you want to waste two IPs on the subnetting? You have to reserve .160 and .176 as network addresses, and .175 and .191 as broadcast addresses. If you don't subnet, you don't have to do this, so .175 and .176 can be hosts.

If you're not using NAT, i.e. if you want to actually do routing and put real servers on those IP address, then you can't subnet your network in a way that is transparent to your provider; they will need to modify their router configuration and their routing tables to account for your new network setup, possibly giving you two gateway addresses and/or two routers (or by setting up a new route if you put one subnet "behind" the other and your firewall in the middle).

Howewer, if you keep using NAT and simply give half of the addresses to a firewall and half of them to another, then their external IPs will appear to your ISP as still belonging to a single subnet, and everything will keep working fine.