The dome temp of the grill and the temperature on the grate differ significantly. In that case, will the S1 temperature be inaccurate?

No. The ProTemp S1 does more than just read dome temp. We use platinum sensors for 10x accuracy over analog, a long stem to get closer to grate level, and a temp compensation function in the app. Our tests show just a 3-5°F difference from a wired grate-level probe in Kamado grill before app compensation.