my suggestion is to put the correct tires on the lift
if any alterations should be made it should be to re-size their door height to allow for clearance (sounds like they didn't spec out the truck well enough before buying it?) I'll guess they bought it for a real deal at some auction? Well saving money is good but sometimes it comes with other costs in the long run for things unforeseen like this.

and of course there is also the option to cut the OHG down but that also has it's costs and you have to get approval from the mfg to do so.

the main thing is to remember that altering a truck away from it's original design is not good, creates all sorts of problems with the truck and possibly for the customer should anything happen because of the changes they make 'out of spec'.....
i wouldn't want to be the one responsible for any accidents because of this and I'm sure you wouldn't either. insurance claims and lawsuits can bankrupt companies, I'm sure they are aware of this :o)

also if this is a cushion tire truck and not pneumatic then they shouldn't be driving it on gravel in the first place, its not designed to do this, that is why they get stuck...
a paved driveway or something is yet another cost they will eventually forego to eliminate the 'getting stuck' scenario. (that is when they get tired of towing the truck out of stuck situations)

imo altering the door to the building would probably be the best way to go and the cheapest since you probably wont find any odd sized tires to fit the needs your looking for.