If you can't get on with your mother, and she can't step back and realise that
she is the grandmother and not the parent, then you have to get your own accommodation, no matter how difficult you think it's going to be.
It's absolutely unfair on your child to live with this confusion and instability.
It's perfectly possible to have grandparents/parents/children living in the one household, but in order for this to be successful you need the adults concerned to be mature, sensible and reasonable.
When my grandson asks me for something I reply along the lines of..."Have you asked your mum? She's in charge"
When he says Mum won't give me/let me etc I reply..."She''s the boss, she decides what happens".
Obviously if I'm at home looking after him and my daughter's out then I make a decision - but always based on what I know/think my daughter would do were she there.
On the very odd occasion my daughter does something I don't necessarily agree with I shut my mouth and mind my own business - because I'm not the parent and obviously unless there was a serious issue of safety then it wouldn't be my place to voice my opinion. Thankfully that's never happened.
You also need to have your own space to retreat to within the home, even if it's just your own bedroom after you child has gone to bed. Turn it into a mini-sitting room with TV etc if you can. You can't be on top of one another all the time.
You also cannot abdicate responsibility for your own child's upbringing/discipline just to teach your mother a lesson. That's just daft.