Need suggestions for good reverse lights

one or two of these (depending on size) should work for that, you could put one on the back and one on each side mounted high if you wanted and not break the bank https://www.superbrightleds.com/cat/vehicle-area-lights/

I have used similar lights on my headache rack from them on the truck for night time area lighting when working out in the field been over 2 years and they are still working and holding up.

Rigid makes some nice led lights but you pay for them. Hard to justify $100+ lights that are going to be on for a few minutes, a few times a year. However their are all kinds of lights out there

I have a pair of high power LED flood lights on the back bumper that I turn on and off by a wireless switch. They're no longer available it seems, and the nock-off look a likes aren't nearly as good. I tried to buy more for other applications and was disappointed.

Anyway, find an off-road light or two that meets your budget and wire it up. The wireless switch is available at many truck parts stores, and also online. You can power them from the bargman "charge line" or off the trailer battery.

A word of caution, white lights that you can control remotely on the back of your trailer are not likely legal. If your trailer has to pass some sort of compulsory inspection, you're probably SOL. I've driven all over US and Canada with extra lights on my trailer and never once been stopped, YMMV.