A power brick with an extended run for power (in addition to
cat5) would be a hassle, and force me to think "Is having a mic worth the headache?"
If your Poe cam needs a separate power source for the internal mic (which frankly, sounds hard to believe- are you sure?), the a Poe splitter sounds like the perfect sol'n
Ie: @ the cam, use the 48v of Poe to create 12v (typical, common) to power a device near the cam. This is what my previous links demonstrated. The Poe splitter, located next to the cam, provides "local power" and means a long run of power, or a brick, isn't necessary.
Post cam make/model of cam, folks here can propose options.
Yes, internal mics don't work as well as external powered mics, in general. Secondly, an external mic can be aimed, but I'm not sure that helps much, since most external mics are omnidirectional. "Aiming" may not help much.
My cheap ext powers mics pick up bird chirps & crickets if volume is set to 50% I hear ppl walk to my front door, and I hear packages dropped at my door (which is far louder than birds, crickets and planes)
So some bench experiments & tests may be in order. And playing on a bench can be fun!