PXE with DHCP on another server

Solution 1:

There is absolutely zero requirement for the DHCP server to be the same machine as the TFTP server (there's no such thing as a "PXE server"). You simply configure the DHCP server to provide the necessary DHCP options for PXE booting. Good luck managing that on a dinky toy consumer router, though.

Solution 2:

subnet 192.168.51.0 netmask 255.255.255.0 {
       range 192.168.51.64 192.168.51.80;
       filename "pxelinux.0";
       next-server 192.168.51.1;
       option routers 192.168.51.1;
}

It looks like the requirement of the extended DHCP "filename" option rules out most router-based DHCP servers. The typical router-based DHCP server is mainly focused around the minimal functionality of just managing IP addressing.