After coming back to this thought experiment a couple of times, I have some rather extreme "solutions" to try to set up a rig which would mask most differences. The first certainty is that it isn't going to be comfortable, in fact it is the specific comfort of each headphone that will have to be masked from tactile perception. The easiest, least elegant solution would be a sort of skin-tight hood which would apply equal pressure to areas around the scalp, coupled with an adjustable weight to mount on top of the headphones. While this isn't a setup that any tester would tolerate for more than a few minutes, it compensates for a couple of important factors: the clamping force between headphones will be equalized, as will their total mass and balance, due to the center of gravity being shifted upwards.
Further, something must be done about the most delicate sensory organ - the ears, with their plethora of nerve endings, cartilaginous flexibility and so on. This is where I propose to borrow a page from tattoo artists and use a mix of topical anesthetics (lidocaine/prilocaine eutectic mixture liquid at body heat) to isolate the epidermis while not affecting the deeper mechanisms of the ear canal (cilia). The temporary numbness will be aided by the sensation of the same weight pressing down on the top of the skull from the weighted phones and hood.
So there's our tester, wearing something that resembles a medieval torture device, blindfolded and partially anesthetized. Admittedly it is an absurd solution but it could be done and would offer some otherwise unobtainable test results. Even then, the value of any results would be compromised by the possibility of interference with the seal/fit of the headphone as well as any psychological component introduced by the not-quite-pleasant physical sensation of numbness and head crushing.
Maybe there's a less insane way to go about this, and if so, I would love to hear some suggestions.