It’s tough to make a parent see your point of view on an issue like this. I think one of the most important things is to be able to pinpoint exactly what it is that your mum doesn’t like about your bunny being inside. Is it the smell of the litter tray? The hay? The mess bunny makes? Things she chews on, or location/size/appearance of the cage? etc. If you can pinpoint exactly what it is, it may give you the opportunity to fix the issue before your mum makes a final decision. Whether it’s trying a different litter or cleaning the tray more regularly, sweeping up hay more often or getting a better hay rack, bunny-proofing things more, moving the cage etc.
Ultimately, if you do have to house her outside, I’d look at upgrading the setup to make sure it’s very secure from cats and foxes etc. Mice are a problem but if you keep pelleted foods to small serves that bunny can finish in 5-10 minutes and sweep up hay regularly mice should be less of an issue. You can also get cat and fox deterrent sprays which you could spray around the area of the hutch (outside of where your rabbit can access, even around the borders of your yard) which may help prevent their intrusion.
Summer heat is certainly a big factor, having the hutch shaded for most of the day is very important, providing frozen water bottles, cool water and a cage fan for breeze can all help. If your area gets very hot, it may be worth negotiating with your mum where if a day is forecast to be a certain temperature or above, then you are allowed to bring your rabbit inside that day, say between the hours of 10am and 5pm which is usually the worst of the heat (depending where you live). That way you can be sure she will never be out on those really hot days, but she is still kept outside like your mum wants.