You can't reliably set the referer in any visiting browser, but you can run them through a proxy that sets/changes referer, and forwards the connection to the target website. You can't reliably set referer to an arbitrary value without any control of the client though.
I do have a workable solution that I can make for you that does exactly what you want, if you can change a single setting on the client browsers. If that's not possible, I can make a proxy solution for you.
I'd normally bid $270 for this, but I can live with $250 to get to within your budget.