Read the instructions that come with the bridge.
I would ALWAYS use straight through with the output of the POE injector. If you had to use a crossover I would use it on the input of the POE injector. But I think a bridge device is just like any other end point device (like a PC) and straight thru would be correct.
How much of a "major concern" can it be? Buy (or make-up) one extra cable ...