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Wheel Balancing - When first try doesn't work
Have them check ream of suspected wheel it's may be bent out of round and most balance equipment wouldn't catch it if damage not to big, it can be fixed. If you have difficulties with new or almost new tire it may be no good and needs to be replaced. As of the size of the tires, you probably driving GM large(performance) sedan and those cars not that reliable as they use to be and wheel bearings is worth checking. If you stuck with one out of balance tire - use it as spare, or at least on the rear axle - less impact on driving this way. And at last, you can make intervention in balancing process, freak out and make them to do it your way. BTW don’t let them use impact wrench to install alloy wheels(to remove it’s OK).