Tillyboo, this must be very hard for you - it always is when in conflict with relatives. But you are so doing the right thing.
My MIL had a dog, which I my SIL and I refused to have around the children. She was a lovely dog, but I once caught doing a 'silent' snap towards one of them. They weren't touched, but she snapped because MIL was paying attention to a child.
MIL couldn't understand the risk would screeched 'she's alright' (meaning the dog). and didn't think it was fair of me to ask her to put the dog either in the garden or kitchen. (wtf).
So SIL and I stopped taking the children round to her house.
But then turned up at my house on a wet day with the bloody dog.
Became very upset with DH as he told her no way is that dog coming in.
- crawling baby
- cream carpets.
so dog went into the garden (and pooed in garden) and all the time 'is sally ok?'
Sadly the dog died - sad because she (without kids in the equation) was a lovely dog, and she was MIL companion (the only thing that could put up with her)
So we started visting her house again. She started talking about getting another dog and said it would have to get on with her cats - and I said children too, and she replied - getting on with cats is more important
anyway, in the end she got a puppy, a lovely little thing, BUT as it keeps jumping up, DD hates it (she's only 17mths) and MIL thinks DD is being VERY silly, we no longer go round. I don't want DD to get a phobia about dogs. And now DD hasn't seen Nana for nearly a month. which is sad as DD appears fond of her and she never sees my parents.
But her safety and peace of mind is much more important.
Sorry - turned out a bit long, but Tillyboo stick to your guns - your LO is much more important than a dog, and I would say can't believe your PILs are like this - but mine are.