It's very well possible that it doesn't fit, but I wouldn't be surprised if they're just being overly cautious. I previously had an f350 (SRW) with a 4" lift and 37's.
Anyway, there were only two out of about a dozen car washes that I could fit into. I used a measuring tape and literally measured the doors and lowest part of the wash mechanism of several car washes to determine which one would work. (Asked permission to do so beforehand. )
My truck was around 87". Most, if not all of the washes around here indicate 84" as a maximum height. Anyway, each time I went to a wash, the attendant would say, "Sir, your truck will not fit." To which I would say, "Do you honestly think I'd be willing to gauge my paint? Trust me, it fits." Point being, they were doing right by assuming it wouldn't fit, but they obviously didn't know whether it would or not. I would literally clear the jets by about 1/2". ( Had to be mindful of any accumulated buildup of sand on the floor during the winter.) As for the hanging bar outside the entry point, when I had taken measurements, I had also asked if I could raise the bar to match my vehicle height. They let me do it. I basically did them a favor, they could guarantee that any vehicle under that height would fit.
I'm not proposing that you go measure the wash, but if you really want to use it, then there's one way to know for sure. Well, two ways, but I don't recommend the second way. Lol