I'd make NO promises when replacing the tires, in fact, for liability purposes I would probably have them sign off that the tires installed will have no traction when traveling on an oily/wet floor.

Trust me- if there's even a hint that your replacement tires will help the situation & then the truck slips & hurts someone you'll be calling your lawyer.

yep, i'll have to agree with johnr
when it comes to oily floors they don't make a non-slip tire
as he said when you mix water and oil together on a tire there's not much you can do about it except pucker up and hope you don't wipe anything out if you start sliding.
Now if you want a more positive traction point you might want to consider doing some work to the floor instead, making it rougher, having traction ridges cut into it or something along those lines.

smooth flat tread with sipping or a walnut shell impregrnated tire will help a bit but not solve traction problem issues when traveling empty on a oil soaked wet concrete floor (most weight is on the rear tires) or turning issues when handling a near capacity load (most of the weight is on the driving tires). Water & oil is slicker than newly frozen ice - close to zero coefficent of friction.

Maybe they need a floor scrubber.