To me, part of the allure of the Route 66 experience is that many refurbished "tourist courts" along the way. Some more famous that others. Places where you might spend the night and have some great photo-ops for the beautiful New Yorker out front of the cabin, for example.
Personally, I have never really gotten into the travel trailer life. To me, it can be another level of complexity added to an already possibly anxiety-producing excursion. Something else to deal with, possibly. Plus something "back there" to worry about and keep track of, as the trip progresses.
Otherwise . . . a good platform receiver hitch, with electric trailer brakes, AND new tires on the trailer. Might consider an external trans fluid cooler for the car, for good measure, plus a fresh ATF change before the trip.
There is also a "Route 66" magazine that is published. Many interesting historical and informative facts in it. Might do some online searching for digital copies!
Personally, I'd rather just take the car, but if y'all want to take your "bed on wheels" with you . . .
Have FUN!
CBODY67