I suspect that the problems with front/back mismatching will only really be an issue if you put crappy Linglongs on one axle and Eagle F1s on the other.
You are looking to leave good grippy tyres on the front axle and put good grippy tyres on the back axle. Unless you are pushing your car to the limits of it's grip (and you come across as a careful driver, Fox, so I doubt this is the case) you are unlikely to notice much difference.
FWIW I have mismatched tyres on my Mondeo, got crappy eco tyres on the rear, which are nearing the end of their life and new Dunlop SP Sport 01 on the front and on this does make a noticeable difference, but only if I'm driving the car hard and throw it into a corner sometimes the back end will come loose while the front is still gripping which I find hilarious in a Mondeo. When I'm driving sensibly it makes no difference.
Yes I'm aware that this is an FWD car we're talking about and that the BMW is RWD.
Go with the Eagle F1s. If you don't like how it handles, then consider buying two more Eagle F1s.