I would not say I'm "unpleasant," not even my OH and children say that. As I have posted before they have said that I, me and BiL are "childish" and need to "grow up"!
The banter works both way by my children and even my OH agrees it was initiated more so by my BiL
Re the dogs name same as mine, my family agree that it is not nice or should not have been done and even my oH has reluctantly admitted it could be to wind me up but its their family and stuck in the middle.
As I said as suggested by the others here and my IOH & kids, ignore, pretend not to be offended and act in moderation that I like their dog.
What winds up my BiL and starts him off is I don't tell him how much the new car/s cost how much we sold them for or the private plate or how much something else cost eG, sofa's wood flooring, hols, etc etc and the reason I don't do that is that is when he asks the question is often proceeded by, and out of the blue as we are talking about something else he would say - "that must be 30 thousand" He has an interest in cars and has an idea and knows that a Mercedes GLS new starts from close to 80k at a discounted price. We did an orangery and he said that must have cost 5k - its 25 feet by 15 so you can work it out
I'm best avoid them but when we next meet I will pretend to get on.
About 5 years ago we sold a car for about 40k, we told him and he banged on about it that we sold it too cheap and then we told him how much we bought the new car for and he banged on about it was too expensive. and should have cost xx ie several k's less - so I challenged him to get us another one of same spec and colour and he fobbed me off that he did not have the time
He is a WUM, more so than me and calling their mutt my name is not on. I will play the game as suggested here and by my family as I know it will wind him up if I don't show my anger.