I assume you mean the fuel filter was replaced when they had the battery out. You said oil
filter which is accessible from underneath the car.
Why would the charge £900 to change a fuel filter, you could have done it yourself in an hour. Just Googling the filter we are talking about £30.
I think you need to change your garage.
I am also surprised that the AA didn't notice the loose battery terminal.
Sounds like you have been taken for a ride. These service advisors should not 2nd guess a mechanics findings, just report what the symptoms you tell him, otherwise it leads the mechanic down the wrong path.
There is no excuse for levels of oils, water and fuel and tyre pressures to be the cause of a problem, these are things that should be checked monthly or if you are a high mileage driver weekly. If you are going on a longer journey than your normal commute then check the car over the day before. Air filter and cabin filter can be checked monthly also, niether takes more than a few minutes and no tools are required. Low pressure or uneven tyre pressures can lead to changes in handling and heavier wear on tyres. Buy a tyre pressure gauge and check them before driving the car when they are cold. Use the air compressor supplied with the car if requiring more air. I run mine at 35 psig and 38 psig if going on long distance runs. If fully loaded you can go up to 40 -44 psig , but check the plate. Also check the pressure in your spare or doughnut.
Its an offence to drive with an empty windscreen washer bottle, if you cannot see the water level then top it up and carry some water in the boot. Don't bother with the plastic dipstic in the cap. Make sure the sprayers work. These jobs are not garage service items but Owners responsibility.
READ YOUR OWNERS MANUAL