FIRZ, off-wheel balancing is sufficient if your wheels and tyres are in good condition. On-wheel balancing is only required if there still is vibration after doing an off-wheel, to take into account other uneven weight factors that come in after mounting the wheels, such as unmatched aftermarket lock-nuts, drive-train issues, and so on.
If you have none of the latter, then it's down to your new 193s and Falkens. Lim Tayar generally know what they're doing. If you bought the 193s new from the BMW dealers, you should go back to them. Similarly, if you bought the Falkens from a reputable tyre shop. If you didn't buy them from the dealers/reputable shop, you could be looking at quality issues.
Another thing is, if you feel the shimmy in your steering wheel, then it's the front wheels; if you feel it in your bum, it's the back wheels. Did the guy say why your back wheels needed remounting? Maybe your front need it too, and if you don't have staggered wheels, you could swop back to front as a simple test.