Hi there,
I'm a Daughter in Law so I'm coming at this from a different angle I'm afraid.
Im speaking from experience as we're in a very similar situation my my husbands parents at the moment, except they don't have any contact with our baby unless we're there as she sees them so infrequently that she doesn't know them at all & would have a serious panic attack if left alone its them.
I doubt if much of what I'm going to say is what you want to hear but I'll say it anyway incase it can in someway provide an insight for you.
Unless she is a totally crazy person (which she may well be but unlikely) there must be a reason she's behaving this way although admittedly a lot of her actions don't sound particularly mature (slamming doors etc). But if I were you I'd be concerned that I'd made her feel that behaving this way was necessary?
I appreciate that it may just be (again I think unlikely) that she's taken a dislike to you but I wonder what she said he you say she spoke rudely to you? What was she rude about?
The main thing that has made me feel compelled to comment on your post is you seem to have put all of the 'blame' onto your DIL, your son is Los in this equation! If they don't visit your house, he surely has a say in this too so why is this purely your DIL's fault?! My MIL is convinced that all of the blame for the breakdown in our situation lies at my door whilst in actual fact my husband is so embarrassed by their behaviour and so angry at them about it that he avoids seeing them and it is me nagging him to visit etc!
As you alluded to in your comments, it is common for the paternal grandparents to be less involved than the maternal side which shows that it is not just your DIL that prefers to see her family more. There is a reason for this and I'm sure that all MILs generically blame the DILs and vice versa but this is definitely a tricky dynamic to manage and you're not alone in being in this predicament.
If you are still able to enjoy your relationship with your granddaughter I would suggest continuing to try to build bridges with your DIL as opposed to leaving her to it. Leaving it will surely only make things more tense between you and if this escalates you may end up being denied access to the baby.
In our situation my PIL have repeatedly behaved appallingly since our little one was born and my husband resents them for this more than I do, we have made o much effort and tried time & time again but with no joy yet my PIL maintain that they have behaved nothing but reasonably and I can assure you that this is not the case at all. This isn't to say that we are wholly innocent, i know there are times that we haven't behaved as nicely as we could have towards them but the difference is that we have done that out if reaction to them and theirs has been constant and often unprovoked. The other difference is that hike we hold our hands up to our part in it, They just can't see that they are at fault in anyway.
I hope you manage to work things out, loving grandparents are so valuable to a family in so many ways x