Is it just me or am I missing something? Couldn't you just write a batch file that you have in the right click menu then it just pass the address to the file that opens the page on the remote PC? I mean you already know the machine name and location of the program...
Run //MyPC/Program Files/Firefox/Firefox.exe "%URL%"
Then you add the batch file to the right-click menu. When it is time to use it, just highlight the code, right click and run the program on the other machine.
EDIT - Just noted that this is what NOSH started to suggest before lining it out. Can someone explain why it would be lined out? I could see if you were trying to make it happen on any PC - you need to target it and verify location of software, etc.; but in this case these things are known. Just "hardcode" them into the target. In fact, for windows, most default browsers are defined in the %path% IIRC.
If not with a batch file, I know we could do it with AutoHotKey. In fact, maybe I will look into making it tonight just for practice. Until then, I am off to bed. Rotating midnight shifts are no fun.