Mum expect us to drop everything so she can see GC

It's OK to say no to her, OP, you just need to practise saying "no" and "not now" and she needs to practise accepting it. The most important thing is what works for you and your baby. Just wait her out if she strops. She'll be back.

she throws a strop and falls out with us

This is ideal. Aim for it. Tantrumming does not get rewarded, tell her that's fine and when she is out of her big feelings she can apologise and you will consider allowing her to visit your child.
It's good practice for when your kid is throwing a tantrum to get their own way.

You're the mum now, start being the mum you want to be, start advocating for the needs of YOUR family, and start saying no when it suits.

Ah that wont work for us (dont apologies, youre not sorry), How aviator X time? Does that work?

If she has a strop just be the parent in the conversation and say "now now, no need for that, of course its not XYZ, do you want to meet at X time or what? Got to go, baby needs a feed"

Jeez. Just say no and if she has a strop let her strop away in her own home!
Why not preempt her by inviting her around at a time that suits you then schedule her next visit before she leaves.

You've got to advocate for yourself and your kiddo. There's no point burning yourself out to keep others warm. I've had the same from my own very keen family and would reply 'it'll be lovely to see you! X will be napping until about Y but we can have a kid-free coffee and catch up while we wait for them to wake up. Maybe you can do some dishes or fold some laundry.....'

Oh come on, set some boundaries or this is going to get worse. So what if she strops? Just say no if something doesn’t suit.

Why cant you say, do pop by by LO is sleeping so youll need to wait until shes up to see her. Or were having lunch so of course but do you mind feeding her. Or even.. no now isnt convenient, come tomorrow at X time?

You need to be advocating for your and your child. If she strows a strop she's only going to end up with less time with your child. She will soon get the message.

youre daft. Say no.

let her sulk. You have the baby she wants to see so she can't ignore you for long! Just pretend you don't even notice and be happy and chatty. She'll soon learn.

If I want to pop in to see my DD or grandchildren, I message her to see if she’s in. I’m pretty certain that if it wasn’t a good time, even if she was home, she’d probably say ‘just out at the moment, will be home at 4’ and I’d reply - Ok, I’ll call round then.
So when your DM calls to say she’s popping round, just tell her you’re out at the moment and will be in at X time.

It’s not mothers, it’s your mother. Why would you rather upset your baby and yourself than your mum? You’re a mother now, stick up for your baby for goodness sake. “Can’t see you this afternoon mum, x is napping”. It’ll get easier the more you do it.