OP, I think this is an opportunity to think long term. Depending on the age of the dog, it could be around for much of your child's early years - a lifespan of 15 is quite reasonable these days particularly for smaller breeds. So, you will need to be thinking of precedents and workarounds for the long haul rather than just this first visit.
YANBU at all to say that you don't want the dog in your home, but your case is not helped if you have previously allowed it. Some of the suggested alternatives (crate, garden/kitchen) may work, or work at certain times of the year, but will be difficult in the winter and once your baby is toddling/walking.
I think you and your DH really need to be very clear between you that if you ban the dog (and your ILs are as, erm, dog-centric as you say) that you will not have the benefit of them visiting you while the dog is alive. If you are happy with that, then that's fine. You will still be able to visit them at their home, and there, the dog has every right to be, subject of course to sensible precautions with your baby as they grow up.
Try again with a conversation with ILs - are they absolutely clear that NO means no, not just well, OK, then, yes, but with a grudgy face? Do you get any sense that there's scope there for some sense from them? I speak as a devoted/besotted dog owner, but I would NEVER take my dogs to a house where they are not explicitly welcome. More worryingly, I am particularly concerned at the severe anxiety shown by this dog - dogs should be able to cope with seperations from their owners such as holidays etc (obviously subject to proper arrangements being made for continuity of care). If this dog can't cope with these necessary separations, and your ILs are unwilling to engage professional help, then I think you will just have to accept not seeing them as much as you would perhaps have wished to.
The only other alternative I can suggest is to research something like a home boarding service or doggy daycare near you - would the ILs be happy with that? Sometimes people are happier knowing their dog is going into a "home" environment rather than a kennel, and quite often there will be other woofs around too, for walks, play etc.
If the dog is really struggling even with that, then the only other option if for them to visit in shifts, MIL seperately to FIL?
Congratulations on the new baby by the way. 