Depends where his height is, all body or legs. I’ve seen 17hh and one 18hh warmblood run barrels without injury but they had excellent flexattion and were very sound long legged, short body, deep chest. Build had quite a bit to do barrels because of the tight turns, sudden starts and sliding stops.
Sounds funny burst they had a half thoroughbred half Belgian that did quite well in barrels for senior riders, nothing to make record books but scores at 18 and 18.5 keeping up with many of the smaller horses (the riders were 70 and up)
Ask the vet what they think about the horses build and the Ferrier about the feet and the legs (kind of second opinion).
Are you going to be riding this horse in a circuit or go fair chasing for competition or just doing ranch fun shows or teaching barrel Racing to younger people
As a learning horse if he’s built for it and enjoys it then you have a great schooling horse, if he shows great aptitude and enjoys it then go for it. While he’s learning though and if he keeps clipping his legs, stumbling, resists the tired etc then don’t force him.
So if he’s built well and sound I don’t see why you couldn’t try him in barrels. Some of the petite smaller horses need to do more strides and even in turns they don’t get to turn as gracefully as a longer stride horse, more strides in a run can also mean more time for it.