We would solve this by configuring our Windows Servers to pipe the printer over to the machine that did the printing.
It's something your IT staff does. The thing is, getting a second printer is far cheaper in most cases.
There's another solution and that is to not use 2 networks but one. There was some false hope that these networks would safeguard infections or block shares but that was blow up years ago.
Your IT should have run down these options for you.
Our office has one internet modem, and connected to that modem are two routers, both with their own network (Lets call them network A and network B). Right now, one of our printers is hardwired to one of the routers (network B). Is there some way to also be able to access this printer from network A?
(*Switching to one router/network is not an option -- we have two routers/networks because some of the work we do involves confidential information, and we are required to use a different network than the rest of the office).