This does sound a long shot…an Audi driver NOT sticking another status symbol on the roof?
I’m pretty sure it will fit, as much as the A4 is an incredibly small boot space, the A3 isn’t as bad (in terms of sizing up against the competition), with seats folded down, no real reason why it wouldn’t fit.
The bike bag will add bulk to the size/shape being packed – why does it need to go in the bag? Would a tarpaulin not be suitable? Buy a large enough one and it will protect all sides of the car as you can fold it over the bike and wheels once it is inside, with the sides also covered.
Bike bag is probably a long-term better idea as you can obviously use the bike to go abroad with your bike, but if it is being used purely to transport the bike in the car, there are cheaper ways…(and yes, I’m not falling for the stereotypical image of an Audi owner where money has no meaning…save cash on less important stuff and you can spend more on the stuff that does count!)