I know it's tough to introduce new things to people with dementia, but what about a case like this? https://www.amazon.com/Sports-Visions-Contact-lens-Cases/dp/B00TOIKQGC/ref=sr_1_37

This way her contacts don't go near the screw top. But again, it's a new concept so it might be more trouble than it's worth, especially because she needs to pay attention to which side is L and R.