Your house, your decision. It's up to you if you feel that a cat will be an asset to your lives for the next 10-15 years, even 20 (some cats live that long). Your daughter may think it's cute, but remember that kids don't realize that kittens grow up to be cats, and that they require work.
You can't force your mother to take allergy medication. I don't know if it's bad for her health or not, but I do know that not all medications work short term for animal allergies, particularly cats. So I'd give up on that.
Your mother can stay in a hotel and your daughter can stay with her there. You can all visit there, go out to dinner, go to the park or the children's museum, see the local sights, whatever would be fun. The cat can stay home alone. If she gets a hotel or B&B, yes, it will be more expensive, but certainly you will have plenty of time to save up for it, just as you did for a very expensive cat.
Pretty much anyone can take care of a cat while you are away - you can hire a neighbor or a local teen or even professional pet sitter to come in, do the necessary work (minimal) and give the cat some attention/interaction. You're right not to get a dog in an apartment with 2 working people, so good for you on that score.
Your daughter can also go visit your mother at her house if you go away. So it costs money to take her there, but you and your fiance can scale back your own vacation costs to allow for that. You'll have to pack up your daughter's toys and clothes, but you would do that if you took her with you. You have to weigh the number of days you'll be inconvenienced (maybe 4 weekends a year? 8 days?) vs. 365 days of pleasure with a pet.
I'm wondering if there's more of a control issue going on here, and less an allergy issue? Does your mother need to know that she "matters" in your life? Does she feel neglected or lonely? Has she been in other family relationships where her needs were discounted? Has she tried to control your choices in other areas? Those could be more underlying issues than what breed of cat you chose.