First post here - funnily enough I have just purchased an F11 528i and am looking at 324M wheels for it as they are quite lightweight and good offsets for the 5 series. Funnily enough I was also looking at E9x M3 wheels at one point too but they aren't really the right offsets for the F11.
That's what bought me here - looking for good pics of them to I can photochop them onto pictures of my car.
However I found this thread and thought I'd comment as I've seen the same things going on with the F8x M3/4 and various F01/F06/10/13 5, 6 and 7 series owners.
With the F82 the rear tyres are quite a bit taller than the front, they have the same side wall ratio front and rear but with wider tyres at the rear. This gave them 799 rotations per mile at the front and only 783 at the rear. When people kept their fronts the same 255/35r19 and changed the stock rear from 275/35r19 to 295/30r19 with the same 799 rotations at the front they had massive problems just like you describe. It was fixable by completely turning traction control off as the actual cause for the issue was the car thinking the rear wheels were spinning under acceleration.
On the E84 with staggered wheels the stock sizes are:
225/45r18 802
255/40r18 799
225/40r19 797
255/40r19 799
So I would have thought that any tyre height plus or minus a few rotations would be fine. Based on your sizes for a few different tyres I searched on they all seem to be about 5 more rotations at the rear:
245/40r18 809
275/35r18 814
I'd perhaps suggest going with a 285/35r18 which tend to be about 804 rotations as then the car will think the rear is going slightly slower than the front however I don't know what that will do to xdrive.
I also wonder if the quite different offset front and rear is creating a weird issue that the steering detects, probably to do with scrub radius - videos here.
I'd suggest perhaps going for a drive without traction control on at all (don't come back to me if you crash though haha!) and see how it goes. That should at least eliminate the chance of the scrub radius being a problem and focus on just the tyre size.
Next I'd try and find some cheap 285/35r18 tyres and see how they go. Are the current tyres all brand new with plenty of tread? If the rears had less tread that'd make is worse but second hand 285/35r18 should be OK and it should be possible to find a few second hand ones for under $100.