I think you have to turn this person away or help them find an alternative place to stay. It's your hotel policy not to accept pets, this was stated online. They don't get to rewrite your hotel rules by writing "bringing dog" in a booking form.
My son is allergic to dogs, his last big reaction had his face break out in hives and his eyes swell shut. Thankfully, I've not seen his allergy affect his breathing, I hope I never see that. So far he doesn't react if I've given him piriton first, and this also stops the reaction (though the swelling took two days to go down).
I only book accommodation that says no dogs are allowed. Recently booked our holiday and had to pass up staying in a gorgeous cottage because they allowed pets. I'd be upset and feel lied to if I found a no pet policy, which is a key factor in where I book to stay, turned out to not be enforced.
Obviously service dogs are an exception to the no pet rules. However there are less service dogs in the world than pets so we're less likely to come across them or stay somewhere they'd recently been. They are also better trained. My FIL has a guide dog and you'd barely know he had it with him...it doesn't sniff people (which is what my son would react to...dog saliva from being sniffed, licked or panted on), it doesn't get on the furniture, she just walks or lays quietly by his side.
A service dog in a hotel lobby or bar probably wouldn't be a problem for us, I would rather not stay in a room which a dog had stayed in recently. Though again, as there's not many of them, it's not as likely so most rooms would probably be fine. I'd bring piriton just in case and ask to be moved rooms if he had any problems.
What I wouldn't be happy with is finding the no pet hotel I'd booked was letting any old dog in. I'd be assuming it wasn't a one off, that it actually wasn't a pet free hotel and wouldn't stay there again. If the dog was a bouncy, sniffy licky type, I would be annoyed that we could no longer relax in the public areas of the hotel for fear of bumping into it.
Hopefully the dog will turn out to be a service dog and not be a problem for you but if it's not, I think you should stick with the hotel policy.