Running a patched tire on a trailer should be fine. Switching it and using it as a spare is fine also, and might be a little easier for your piece of mind.
I've run "plug" patched tires for decades - on cars, trucks, and trailers - and never had a problem. I have done my own plugging, and have had tire shops do it.
Last time I asked my local repair garage about an umbrella type patch he said he could order one for me. He stopped carrying those years (decades) earlier because he simply never had problems with plug patching.
As to a hole / leak in the sidewall of a tire... every tire shop / garage that I've been to refuses to repair a hole like that.