What would your dh say if you said that he needed to call your mum 'mum' too?
Obviously you don't want him to do this - he has his own mum just as you have yours but it would be interesting to see if 'I don't see what the problem is' would translate to him being actually happy to do this when asked...
Does your mil call her mil mum too or does she use her name?
Sometimes if she gets upset by things that upset you then you just have to show that you are more upset that she is - every time she suggests it, just have a few stock phrases to keep repeating until it sinks in - 'I thought we had discussed this. I already have a mum. You are not my mum, you are my mother in law, in my family it would be incredibly disrespectful to my mother to call anybody else mum. I can't believe that you are pressuring me to be so horrible. You're not respecting me or my mother when you ask this'.
My mother used to call her mil 'mum' which she didn't really like but sort of got guilted into it from what I can understand.
My dad - don't remember him calling his PIL mum and dad - but when I was a baby I couldn't say their names properly so they had nicknames that stuck - and he used those. As did my other cousins and aunts that married into the family, as did family friends that knew them through our family, or if my friends were there at the same time...
I managed to get away without calling my mil anything in particular - didn't really speak to her a lot, definitely didn't want to call her mum (although some of the other siblings' partners did) and felt odd using her first name too.
So it's one of those things that's not unheard of but not normal either. Suspect there's more regional or cultural or family influences that come into play than being a definitely right or wrong thing.
But if you don't feel comfortable calling her mum, then don't. And just keep on reminding her with the 'we've discussed this. I'm not calling you mum, it's incredibly disrespectful etc etc'. If she doesn't want you calling her by her first name, then maybe there's a different word that you could settle upon that is a compromise - maybe mum/mil/granny in a different language that one of you has ties to, (if there's any welsh in you there's a word for nanny or granny that when I was little I could have sworn was monkey - I think it's something like mamghi - but maybe offering to call her monkey would help!), or a made up word or something else that you could agree on? Then at least you would be keeping mum for your mum.