AFAIK, the 10-speed has no provision to add auxiliary cooling.There may be a requirement for a tranny cooler for trailering.
Agree with your caution about renting a trailer. Also, check it very carefully before heading out on the highway. I once rented one that had a wobbly wheel due to a completely annihilated wheel bearing.
That aside, a trailer has the advantages over the other options of more space and better balance.
OP, you mentioned a 4x8 trailer. The national rental company that I often use has many sizes. Get the smallest one that will hold your gear. This will minimize your total weight. (The 4x8 from a certain rental company weighs a hefty 850 pounds empty.)