Suggest that bar tape may be a secondary concern. Was the frame size and fit determined by a professional fitting session? Has the rider had a medical examination to evaluate hand / wrist conditions?

Bar tape can lessen some road vibration, but if there are already issues while wearing padded gloves then more likely too much weight is being put on the wrists or there is a medical condition that makes them overly sensitive.

A higher riding position up front would be more effective than adding another layer of padding. That could mean raising the bars, going to a shorter stem or changing the bars altogether. And that assumes that the frame size itself is correct.