Fun idea! I think the difficulty should scale a bit faster though. Maybe the scales should tip more easily relative to the weight difference. Also I think dumping pieces off the sides to not add excess weight is too powerful.
Perhaps the game should punish you for dumping pieces too frequently? Raising the side barriers could have also made that strategy less viable.
Lastly, making the pieces slowly fall automatically like in Tetris to give you less time to figure out where to place blocks could have added additional pressure to make decisions quickly and speed up the game.
Overall, awesome idea and polished execution, just needed a couple things to spice it up difficulty wise :)