And just to mention, the only 'feelings being hurt' here are your DPs. He's upset because you're not doing as you're told, by him.
There is no suggestion in what you've written that his DB is upset, or even bothered one way or another, whether he can bring the dog. Why would he have strong feelings, become emotional, about that, when he has somewhere else the dog can go? Is he weirdly emotionally incontinent generally? Have you observed this for yourself?
Surely to him it's just a simple practical arrangement, one way or the other. He just needs to know yes or no, so he can get on with it.
The only person 'getting upset' here is the messenger, your DP.
He wants to be the big man who can invite whoever he likes to your event, play host to his guests, at your party. Because he's more important than you and your DCs, so his wants have to come first, every time, whatever the event. He's putting his want, for status, far ahead of your wants, your DCs' needs and their guests' needs.
Text the brother. Take away your DPs power to control this situation. It's likely the brother will be fine but your DP will tell you he was upset. If the brother does get upset or angry with you, you'll know they're two apples who've fallen not far from the same dysfunctional family tree.