Of course you're within your rights. I think the first thing to say here is - if she is known for being loud and domineering - then use this 'shift' in the balance (ie DD arriving and you and DH becoming 'the parents') to draw boundaries. It's new territory - colonise it first. It'll be scary, but do some putting your foot down, to demonstrate that yes, you WILL be the one making the decisions here.
Note - this isn't so that you can push MIL out, but rather the opposite - so that you can be happy letting her in. This goes double as your DH is also not exactly her biggest fan, by the sound of it. If you wimp out and let her start bossing about with your DD, pretty soon you are both going to not want her around - and that would be a shame for her and DD. If there can be a good relationship with granny, that's fab. There can't be a good relationship with granny if granny tries to overrule the parents and isn't shown where the boundary is - eventually, you'll fall out. So, if you see what I mean, it's kinder to MIL to make it clear to her that you make the rules and no, you're not going to let her be in the driving seat.
So - nursery. I'd say no, simply because it doesn't sound like a good plan at all. Your DD doesn't know her. I don't know how old she is, but it sounds like the plan is that MIL has DD alone for the day while you work, instead of nursery? I can't imagine a better way to put her off MIL for life
- she gets introduced to this stranger then left with her for the day, in her own home instead of nursery? - and a stranger who (the minute you are out of the door) is probably going to be all over her with the hugs and kisses and granny loves you? I imagine DD might spend a lot of the day pretty freaked out and crying for you or your DH, and that's not going to be great for MIL.
I'd explain this, and say no, but try and work out a way in which MIL can get to spend as much quality time and getting to know DD time as possible during the visit, to build their bond.
And if MIL starts to disagree with what you say, you smile and say 'Nope. My child, my decision.'