As I mentioned above, if I continuously ping the router from an machine
behind the bridge, the IP address that the ping is coming from is able
to receive data properly. Machines on the router side of the bridge
are able to initiate connections to it.

after you kill the continuous ping from a machine behind the bridge, it
takes anywhere from 10 to 40 seconds for the router to "forget" how to
route packets to that machine. So the number seems to vary a bit.

If I ping the router with an interval of 10 seconds, then the router
seems to never lose the routing information.

Talk about an ugly workaround. Ugh.