God this board needs a resident legal adviser ...
That being said, in the eyes of the law, animals are property. If this was a divorce situation, yeah, you could sue for visitation. It's not.
Trust me on this one, there's plenty of homeless cats out there and in a few months there's gonna be TONS of kittens swamping the shelters. 95% of all cats are um, kind of alike (dogs, too). So if you really loved the kittens I assure you that you will fall in love with new ones. The odds that the kittens you owned of falling into the unique category are extremely slim -- seriously, 40 years I've owned cats, completely lost count of how many I've had over the years but we're talking way up there, and I've only owned maybe five truly special and unique ones.
So mourn for a bit then go to a shelter and adopt a dog/s or some kittens, it's all good.