Are the tiles the same cost per sq ft? What waterproofing system is your contractor using behind those tiles?

Did you ask your contractor the reason for the extra cost?
That might be a start : ) I kinda thought the price sounded fairly reasonable--is he doing all of the install as well as the tiles? I've priced a lot of tile jobs a zillion different ways, also done a number of my own. The labour/materiel cost savings from 4x4 to 6x6 tiles that you're citing (cutting fewer tiles, less grout, fewer tiles to buy & set) really are negligible, probably in the range of $20-40, if that--depends on the size of the job.

Sometimes larger/thicker tiles are harder to cut, requiring a wet saw instead of a tile cutter which will add to the cost, or they might break more easily.

If you're looking to save money, skip the tiles and go with a pre-fab shower unit.

Larger tiles doesn't mean any less cuts. In many cases, it means more cuts because the smaller ones can fit into the space without major cutting. And more wastage needs to be calculated into the mix. 6x6 also costs more per square footage for the tile itself.

But, you're focusing on the cosmetic, and the important thing is how the shower pan will be built and how they will waterproof the walls. You need to find out about that or you may be doing the same project again in 4-5 years with leaks behind the walls.