Will it be more feasible to order the turf with no shock pad if I'm pairing it with 3/8 thick rubber tiles so that it won't protrude when installing all of the flooring for the gym?

The thickness of the turf with out the shock pad is approximately 3/8," however you'll want to keep in mind that the grass will compress whereas the rubber will be a solid surface. We suggest ordering samples of each product to try out your preferred arrangement.