the open trailer has a 4 way?
Ill tell you what i did with my mower trailer years ago...
I built a 5x10 from the ground up. at the time "leds" werent in existence for trailer lights.
I have 2 self contained oval strobes surface mounted in the fenders... they are wired to a switch near the tongue, the switch is powered via the park light circuit... so they will come on only when the vehicle lights are on. I also connected that switch's power wire with spade connectors. the reason for this if there were ever a need to run the strobes without a vehicle attached i could attach a small externally battery on the trailer and hook the power from the switch to it and then attach a ground wire via an alligator clip or something to the trailer....
you could do the same with yours...
now for your enclosed trailer,
I am not familiar with 7 pin connectors but i believe there is a constant on with those?
you would need to trace that wire, attach the power to your led flasher to that wire and then your leds to it...
other alternatives... small AGM battery and a solar panel with charge controller. (found at most "big box stores")
how long are you planning on running these lights?