I think PILs have tried to offer a compromise to both parties, with the baby gate so dog can be kept away from SIL.
I think it's now up to SIL to decide if she wishes to accept this compromise herself, or chose not to attend at all.
It's very unfair in any situation when one party tries to compromise and the other refuses and therefore gets their own way because they are one who refuses to budge at all.
It great resentment with those who were prepeared to make some concessions but find themselves forced into going along with the preferences of the intransitory person.
This often happens in families where there is one person who views their needs as greater then everyone elses and the more easy going people make efforts to placate them but resent develops over the years you realise 'why I'm I the one who has to give in always, why are my preferences less important?'
This isn't just about dogs it's about compromise.
I think some efforts need to be made to accommodate her needs, such as Those that have been offered: a baby gate and great care taken to keep dog and her in separate rooms, she now needs to make some accommodation for PILs efforts.
OP needs to accept she doesn't like dogs and keep dog away from her, SIL needs to accept family do like dogs and sometimes the dog will be present in the house when she is there.
Not liking dogs does not trump liking them, or vice versa. No one is right they both have to compromise. A compromise has been offered by PILs.
Ball in SILs court.