We are running into a similar problem, and will probably end up using something like
this. It is a 4 port switch not much bigger than the router, and available for less than $10. It can be powered off the VRM. A splitter will not work, as you need one on each end of the cable. They take advantage of the fact that 10/100 ethernet cables have 8 wires, even though they only need 4. They basically send two ethernet connections through one cable, as I understand it. So you still need two output ports at the end.