Googled pet motorcycle trailers and found this: https://pbmotorcycletrailer.com/pet-...ycle-trailers/
Check with your trike mfg. about max trailer weight specs. Remember towing vehicle weight should be substantially more than towed vehicle for safe operation. Another thing to consider is the speed rating of the trailer tires. most small diameter trailer tires are only rated for 62 mph max. It's easy to get rolling 70 - 80 mph on today's interstates.
I'm thinking that you could build something really light using some 17 - 19" dirt bike wheels and tires speed rated at 80 - 100 mph for an added safety factor. You could also use the dirt bike spring/shocks for the suspension also.