I'm renting a Fiat Ducato (5998 mm long, 10 m^3 payload volume), that according to the rental company is 2640 mm high, but, according to the FIAT specifications I could find, the only compatible value I found is 2520 mm when unloaded (and 13 m^3 payload).
A garage hallway where I have to drive is 2450 mm high in a short part of it with transverse beams. When loaded with 1200-1400 kg there are no problems, the van will surely fit. After unloading, it won't fit anymore.
How much can I gain by releasing pressure from the tires? on the way back only three people will be inside the van.
For information, after the garage I only have to drive down a ramp and few hundred meters to get to the closest place where I can pump the tires back to nominal pressure.