Try 2 layers of reinforced tape, one wrapped (as tight as possible) clockwise, the other anti-clockwise, with a 25-50% overlap. At most the shaft width will be increased by the thickness of the tape x8 which is probably 1mm overall. Shafts vary from 6-7.5mm and those aero differences are negligible. The further the air resistance is from the fulcrum, the greater the effects felt will be. Therefore the top of the head being thicker will have more of an effect than the shaft.
It is essential that the adhesive has to be able to withstand the flex plus the tape be able to stretch and compress a little. In reality the shaft is only flexing a small amount, it's not like you are trying to go from a shoelace to a solid steel rod. It's simply about limiting some of the already small amount of flex.
Oh, screen protector tape is the same or similar to "helicopter tape" but it can be thick and might be too stretchy.