Ok, so I went to this new shop and, as expected, one of the wheels did have an excessive road force detected.
The technician said it's the tire that is to blame. Even though I really pushed him to take the tire off and check the rim properly he assured me that it's the tire. A new Michelin tire... very disappointing.
He managed to get it repositioned on the rim, from the second attempt, in such a way that there was no more warning on the road force machine. The wheel was mounted on the rear.
They rebalanced all 4 wheels and I'd say it's now 99% smooth at 87mph. I am already super sensitive because of this whole nightmare so it seems like I feel every single shit and probably even imagine some.
The best thing is that vibrations don't increase even if I take it up to 120mph, which I did briefly, just for a test. I normally never exceed 95mph, in short bursts, to overtake someone.
I was thinking afterwards about next winter, when the bad tire will have to be moved to the front. So I called Michelin and they said I can take it to them in spring to have it checked out. If it turns out to be out of spec, they will replace it. If that is the case I'll probably need to buy a new one as well to pair it but at least there will be no vibrations.
Thank you all for your inputs! And thank you
@stoopid for teaching me about road force. This is where I'll do everything from now on.
PS: If an admin could change the name of the topic to something less dramatic, maybe it will be of use to others in the future.
Perhaps: Successfully balancing your wheels in less than 7 attempts.