My MiL tried 'love bombing' me last year when she realised she hadn't seen me for three years. She unexpectedly sent me a gift when I was in hospital. I responded with a thank you via snail mail - no way is she having my personal contact number.
I find it quite amusing that the last time she saw was me a couple of months after I'd nearly died in an accident, then again from surgery complications. Not only did she not bother to ask how I was (which I didn't expect) but she cut me dead and refused to speak to me in my own home (which I did). A brief 'hello' would have sufficed and is the most basic courtesy anyone is entitled to expect.
By this point, I was through. It was the last time I saw her and the last time I intend to. Long story, of course.
As to your MiL, I'd say from personal experience that the best way to deal with this is to withdraw. The 'push pull' idea you describe is a common part of the cycle of emotional abuse: you can Google several variations on this theme. Without the occasional kind parts no one would tolerate their behaviour, so they give just enough to keep you embroiled within the cycle. I have no time for people who behave like this. I'd drop the rope and let your DH contact her in future. She is his mother, after all.
And sorry, but this 'joint custory of a dog' thing is over-emotional, anthropomorphic BS. Dogs are not children, do not require the same care as children, and are not at the same level of priority as children. I've always been fond of dogs, which I treat as dogs, not as 'fur babies'. Some owners these days have completely lost the plot and fail to recognise that treating dogs as human beings is not necessarily in their best interests.
Protect yourself at a distance, OP. 💐