You should be able to. Should mind you. Only reason I think this is the case is because I have a Fleet Assault Cruiser, and at one time owned a Mirror Universe Assault Cruiser, and the mirror skin is available for my Imperial.
Edit: Needs to be same ship, thank you Mando for pointing that out. So if you have the Mirror Universe Advanced Escort, then you should be able to use the mirror skin for all the other Patrol Escort models.
How exactly do these skins work with the ships they are copies of?
I.e I bought the Mirror Patrol Escort. My understanding is I can use its skin on the regular old Patrol Escort.
Can I use it on the Fleet Patrol Escort? That would be cool.
The Mirror Patrol Escort should unlock the skin on the Advanced Escort (as well as the MVAE and Fleet Advanced), since they are all the same ship model (Prometheus and variants). To get the Mirror material on a Fleet Patrol Escort, you need to purchase a Mirror Advanced Escort.
So if I buy the Fleet Patrol Escort I can make it look like the Prometheus if I have both the Mirror Escorts?
No, Fleet Patrol will only look like the patrol escorts. To get the mirror skin for it, you need to at some point in time have owned the Mirror Advanced. The mirror option will stay even if you dismiss the mirror ship.
For example, my Fleet Assault Refit has a mirror skin option even though I dismissed my Mirror Star Cruiser ages ago.