She has a very short temper. My dad and I have been walking on eggshells for years. - so she's got a nasty temper, is controlling and has potential to outbursts. Doesn't like her other gc because they don't put up with it. LIkes your daughter because she's laid back - AKA lets her do what she wants, without question, and without her own input or wishes being taken into account - sound like anyone, OP?
You already don't include her in stuff with your InLaws, as you know she will either kick off, or she will be embarrassing and judging and will ruin the day.
So basically, she likes your daughter, and occasionally, you, if you do as you are told. How can you submit your daughter to this? Don't you think your daughter should be being taught how to draw boundaries, to make her own decisions, to stand by her own decisions, to have her wishes taken seriously? Your 'D'M will cow and break your dd in the same way she has you - and you are willing to allow this?
YOUR MOTHER IS NOT A NICE PERSON! She has made you think she is, and we are conditioned to believe our mothers are saints and martyrs and would do anything for us; but some aren't. Some are controlling and manipulative and vicious, and should be exposed for it.
Seriously, take a huge step back. Look at other 'normal' families, like your ILs. Do some reading on toxic parents. Check out the Stately Homes thread.
The scales will fall.