I personally won’t put under the care of any parents if it requires them to quit their job. This will be the thing they will be saying every single time (I quit my job to take care of your baby bla bla). Put aside money, I think safetiness of my baby is more important (I’m not worried about the physical scratches by accessories, but more of what if any Jewelry gets loose and baby swallows it?) It’s worrying enough to be away from baby, let alone constantly having to worry about baby getting hurt or possibly in danger while I’m away at work.
For allowance wise, I wouldn’t say your mum is wrong for quoting her price. She have all the rights to quote her price for her service, so there’s no fairness or what here (same thing, you outsource home based nanny, they have their own rates).
I would personally put baby in IFC and just give allowance (whatever I can afford without putting myself in financial crisis). And don’t forget, 2 elderly woman together with a helper, nothing goes well. You’ll get all the “she do this wrong, she do that not good” every once in awhile.
Also don’t forget if she were to quit her job just to take care of your baby, you have to do things her way. If you complain, you’re ungrateful and you may risk losing out an IFC spot (dragon year damn hard to get spot) and also a caregiver which then risk your job.
Keep the relationship clear and try not to involve money if possible.