Looks like fun! I would say that if the tread is packed firmly enough that the dog's paws are staying on the surface, you're okay. I wouldn't be surprised if you encountered other riders who thought you shouldn't be there, so I'd be prepared to calmly explain that the dog isn't hurting the tread and hopefully they accept that, but people can be irrational. Someplace like Quarry Park that isn't officially groomed (and is frequented by plenty of dogs) is perhaps less likely to result in confrontation. Though it may be difficult to navigate through those tight trails.
