It's normal for a BMW to have uneven wheel gap on the left and right sides of the car.
The only reason I can think of why your car would pull more when weight is added is increased camber and toe. This is inherent to our suspension design - the camber and toe increase as the car gets lower to the ground. But I think that's not the actual problem - something bigger is at play (e.g. something is broken/worn on one side of the car and letting the toe/camber go further out than usual).
-How many miles on the suspension?
-Has any suspension component been replaced?
-Could you post the alignment sheet?
I would get the car's suspension inspected by an independent shop used to working on BMWs.
2011 BMW E92 335i, 6MT, Le Mans blue - Quaife LSD, JRZ RS Pro, JRZ camber plates, Eibach ERS, full M3 susp, custom toe arms, Powergrid end-links, SS brake lines, brake ducting, Michelin PSS, UUC SSK, ETS FMIC, ER CP, Perf. Exhaust, VAC RSIK, Sparcos; work by VAC Motorsports in Philadelphia