Agree with the other comments on the trail busyness. I’ve done this a few times towing my kids; it’s definitely doable, but you’ll have to be patient as it is narrow and the peds/joggers always seem oblivious to the possibility of cyclists on the trail needing to pass.
Only thing I would add is that there is a bike rack just inside the main zoo entrance, up the hill a couple hundred yards (across from the park police station) off of Beach Dr. The trail takes you right there, but first you’ll pass by a service entrance to the zoo on your left. That (the bike rack) is where I’ve locked my bike/trailer when towing the kids.
It’s probably also worth mentioning that Beach Drive (starting a bit north of the zoo, at Broad Branch Rd. https://www.google.com/maps/place/38%C2%B056’39.5%22N+77%C2%B002’59.6%22W/@38.9442981,-77.0515083,17z/data=!3m1!4b1!4m2!3m1!1s0x0:0x0) is closed to car traffic on weekends and holidays and is a very pleasant place to ride.