You can run your non-registered dog at a HRC regular hunt test IF your rescue is a pure bred dog. Mixed breeds are NOT allowed to run in HRC test regardless of UKC registration.
At least one other thing to keep in mind:
Clubs are mandated (by UKC/HRC rules) to give preference to all UKC registered dogs over all non-UKC registered dogs. This is true even if your premium arrives before a premium for a UKC registered dog. UKC dogs have preference over your dog and your money would then be refunded if there are no more spaces at the level you want to run your dog.
Just wanted to say I think it's great that you are training and running a rescue dog. I wish there was a program out there that specialized in rescuing sporting breeds, working with them and trying to find them hunting homes!