Nick
this may well have nothing to do with balance….
As trail rat correctly mentioned, as well as variances in weight distribution, there can be variances in shape (tyres & rims not perfectly round) but also (in this case and more likely based on what you wrote) a variance in the uniformity of the tyres spring rate.
A tyre is made up of flat components, that are subsequently ‘rolled’ up into a circle and where they join or splice,and there can be a stiffer segment due to the necessary overlap.
It is not realistic for manufacturers to measure 100% of tyres for this in the factory hence there are very occasional market issues identifiable by a vibration that cannot be balanced out. Normally this vibration is present at a lower speed than a (slightly) out of balance tyre would exhibit.
This uniformity cannot be seen of course nor measured with a regular balancing machine. There are however some machines such as the Hunter GSP9000 that have a feature that allows it to be measured.
Its a frustrating thing for the dealer also, as he will need to claim to the manufacturer for the tyres and its not something that can be definitively proven without a special machine.
(ps: I am a tyre engineer by profession)