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Personally, I think this problem is ill-posed. What geometric properties would the OP like to assume about $S$? What does "constant" mean -- constant on the set of rational points, or truly a constant polynomial? Also, what precisely does $d$ "significantly smaller" than $q$ mean? There are "surfaces" in $\mathbb{F}_q^3$ that have few rational points, e.g., the vanishing set of $x^2-x$ is a surface with only $q^2$ points. Worse yet, a polynomial like $P(x,y,z) = [(x-1)^2-1]^2 = x^2(x-2)^2$ will vanish on all tangent vectors at these $q^2$ points. Of course $S$ is reducible, but the OP says nothing about irreducibility. I suspect that there are similar irreducible examples: the crucial point is that the (few) rational points (mostly) lie on a small number of curves cut out by low degree equations.