I would solder and and heat shrink the connections. Samller heatshrink for the wires is easy to find. May have to look at an electrical place for large heat shrink. Solder will give you a better electrical connection. The heat shrink should keep the connection dry. And, it looks a lot better than tape.
Most butt splices or scotch locks are a mechanical connection; they are prone to corrosion and mechanical issues(how tight the crimp is ect). Trailers have anough wiring and lighting related issues.
And, it would look really like a professionally done setup. That is, if anyone can actually see the splice... You can not tell where I did mine unless you look really close.
+1 solder them. Stagger the connections so that you have a 6" +/- section of fat "wire bundle" roughly the same size from the soldered connections, heat shrink each wire individually, then wrap with electrical tape or put inside one of those plastic wire loom covers.