I know that this response is a bit late but one way you all can handle the no pet issue is to say the following in your policies.
"We do not allow guest pets anywhere on our property including our parking lot."
This would mean the guest couldn't bring them in the house or have them in their car on your parking lot.
If they chose to park on a public street, that is out of your hands.
Unfortunately, guests love to take advantage of the good nature of innkeepers. They get upset when we stand pat and enforce our rules. Threatening and innkeeper, just makes us more resolved to enforce the policies. It is just a pain that we have to deal with guests on TA that do lie. The best policy is to always document what happened right away in their reservation so that you don't have any lapse in memory of what went on..