OP, do NOT Give up your bedroom. YOU are the adult and YOU need space as well. Why should an adult give up their only space so a child can get what they want when really all a bedroom really needs to be is somewhere to sleep in?
Plus the OP has spine problems so should her health be put at risk just so a teenager gets their own way?
Children at some point in their lives have to learn that life is not fair and they cannot get what they want every single time. Not every person can afford a bedroom for each person, such is life.
Millions of children have had to share a bedroom with much older or younger siblings and as one of them I turned out pretty damn fine.
Best bet is to split the room in half as others have suggested (which my parents did once my older siblings had moved out so myself and my younger brother could have our own space. Dad and brother put up a stud wall with a small corridor and 2 temp doors. We both started off with bunk beds (so friends could stay over) but was just brill!)
And children do not need their own bloody bedroom for studying. Believe it or not people you just need a quiet corner of the house to do this, not some bloody dedicated room. She can do this in the bedroom/ mums room/ kitchen or living room, wherever is the quietest. Its not like the house is over run with people, there will always be a quiet place to study in there.
As for privacy well you find a private space anywhere really. on the sofa listening to music or reading/ sitting at the kitchen table/ even lying on a bean bag/ floor. I ever much doubt a 3 year old spends all their time in the bedroom anyway.. they just sleep in it so she will have plenty of "private time" in the bedroom until it's the little ones bedtime.
Even when she is older its going to be a lot easier with a much younger sibling that one nearer her own age. At 5 her little sister will not be borrowing her clothes/ make up/ having screaming matches with her for losing an earring... just might ask her to play with her every now and then. :)
And she doesn't need a special dedicated outdoor space either. What is she going to be like when she leaves home and finds out that she has to flat share and she has only a tiny room to call her own? Will the OP have to go and build her a special "outdoor space" for her as well?
As the the not thinking 5k is much money.. does she have no concept of money and prices? then you need to sit her little bottom down and show her the real cost of living. Do a spreadsheet of what comes in and goes out.
Then get a nice big red pen and strike through all the things you will not be able to afford just so you can get a bigger house so she can have her own room.
No new coats/ no holidays/ no birthday or xmas presents/ no pain meds for when mums back gets bad/ none of the food she likes/ no games/ no mobile phone/ no internet or netflix..... plus she won't be able to have any of her friends over to stay in her new room as you can't now afford a car and you will have to move 20 miles away just to afford a 3 bedroom property and she would have to move schools etc etc etc...
My mum and dad did this when my middle sister started to act up and complain about not getting what she wanted and wanting more pocket money, she soon shut up and even went and got a paper round.