This doesn't add up. Surely the black bin lives outside in the garden/drive? If I'm throwing things away inside the house, I use another bin or bag to put them in first. Just for the practicality of getting them to the outside bin. So my black bin contains a number of bags as it nears collection date. Most are bags from the kitchen bin. Perhaps another bag from the bathroom bin or if I've been having a particular clear out. I'd very rarely throw lose things into the black bin.
Maybe a few non -food/household items if I was passing, or clearing out the car, but they wouldn't be in the bagged kitchen waste, they'd be popped on top of or between those bags. Or more likely in a bag of their own if I'd done an in room clear-up. I think there's an excellent chance tbat your make-up bag is nestled among non-food waste, not in the main kitchen waste bags and absolutely retrieveable.
If your mother has genuinely thrown this away in a cleaning sweep, she should be able to tell you how she binned it - into an existing household bin or bag before the main black bin and what that looks like?
If she's vague or just 'can't remember' , I think she either has cognitive issues developing or has binned the make-up bag deliberately to upset you. Or perhaps is saying she has binned it to upset you, when really she's put it to one side. Or has she put it to one side but can't remember where and is worried about saying that?
Either way I'd pop some plastic gloves on, lift the kitchen waste bags out of the bin and see if it's lying between them. Or if there's a random plastic bag in there with just household items in it? If it is in the kitchen waste bags, they are normally quite thin and you can probably see or feel the make-up bag through them and work out whether it's contaminated before you open the bag.
If you really can't face doing this (I've had to a couple of times and it's never as bad as expected), then yes, please show her an itemised list with links and prices and say you'd value her help with replacements.
Otherwise there's no incentive for her not to do it again. I wouldn't ask for replacements unless she can easily afford them, which from your account she can.