Sorry I am a bit late getting to this but just saw it. I have designed and sold rack for over 30 years both in the UK and USA. 6" Clearance between top of pallet and underside of beam is standard. I would go down to 4" if it was a slow operation or there was some kind of vertical selector.
To do it on less than this is ridiculous and totally unsafe. Especially with a product that is A. breakable and B.? of high value?
I would also venture that the Warehouse Management are violating U.K. Health and Safety regulations by requiring that drivers operate in an unsafe manner that endangers them.
Report them.