Hi Nick,
The tiles used in car showrooms - most of not all are a matt finish and not Glossy.
Once the floor is clean and free from debris spray buff the floor. Use a WHITE pad.
Use something like Mop & shine from Carefree. But for it to shine it must be 100% clean.
After floor has been buffed you could give it an extra buff with a carpet bonnet or two depends how big the floors are.
We have done lots of car showrooms this way - They all were impressed with the results, but then you would need to clean the grout as well if you wanted it superb.
I think if you did this you might open a can of worms though as they may want it like that all the time and higher standards take time and effort which you may not have priced for if it is a regular contract.
Dave