Can you ping the printer during normal operation? Sorry, what do you mean by that? Can you ping the printer from a client when it is in standby?
From most of your answers the operation is ok..
All network printers must have an IP address in the network.. I set my printer at a static address and I can ping it at any time.
The printer has not been in use for 12-24hours.. and is in standby.. but ping must alway be responsive. So open a terminal on a computer and ping the IP..
ping 192.168.2.35
PING 192.168.2.35 (192.168.2.35): 56 data bytes
64 bytes from 192.168.2.35: icmp_seq=0 ttl=64 time=9.799 ms
64 bytes from 192.168.2.35: icmp_seq=1 ttl=64 time=8.345 ms
64 bytes from 192.168.2.35: icmp_seq=2 ttl=64 time=7.171 ms
64 bytes from 192.168.2.35: icmp_seq=3 ttl=64 time=8.355 ms
64 bytes from 192.168.2.35: icmp_seq=4 ttl=64 time=7.156 ms
^C
--- 192.168.2.35 ping statistics ---
5 packets transmitted, 5 packets received, 0.0% packet loss
round-trip min/avg/max/stddev = 7.156/8.165/9.799/0.974 ms
it is possible also the issue is DNS so ping the printer name.
So when I ping the printer by name (must use the full domain).. then it works.
If I attempt to ping without domain it fails.
ping lj2300
ping: cannot resolve lj2300: Unknown host
ping lj2300.local
PING lj2300.local (192.168.2.35): 56 data bytes
64 bytes from 192.168.2.35: icmp_seq=0 ttl=64 time=9.478 ms
64 bytes from 192.168.2.35: icmp_seq=1 ttl=64 time=7.461 ms
64 bytes from 192.168.2.35: icmp_seq=2 ttl=64 time=7.423 ms
^C
--- lj2300.local ping statistics ---
3 packets transmitted, 3 packets received, 0.0% packet loss
round-trip min/avg/max/stddev = 7.423/8.121/9.478/0.960 ms
The printer must respond during standby in order to wake up and print.
I wonder if you have made such a large change to the network, that your previous experience is even relevant.
Yes, we reinstalled the printer, even installed a New one.
If you installed a new printer was it the same as the old one?? Were you trying to get around the problem of not able to print in standby?
One easy way around this is to change the standby period to much longer time.. if you have a lot of people in a network printing make sure the printer doesn't sleep..
I also wonder if the printer setup has actually changed more than you think.
When you had the printer setup with the non-apple router.. the print protocol being used was probably different to what is used now with apple router. So it might be a case of how well does the printer handle bonjour.
Apple networking still has mysteries for me.. but I recently discovered something incredibly useful.
Safari has the ability to check your bonjour network. (You might find it has to be turned on in Safari preferences).

Check this out when the printer is in standby.. and when it is in normal operation and see if you can stop the printer disappearing.
So much will depend on the actual printer and its capabilities. It is difficult to give you hard and fast rules.
Also bonjour itself has issues.. there is a sleep proxy setting that is causing trouble. So read up to understand what is going on.
About Wake on Demand and Bonjour Sleep Proxy - Apple Support
Try and work out what protocol everything is using to the printer and how that might have changed with the change of routers.