What is the expected number of different sequences of bits in the collection that contains $2^{800}$ elements?
Every input $G(i) \mathbin\| S$ to $P$ is distinct.  If $P$ were a uniform random permutation, the random function $x \mapsto F(P(x))$ would have almost uniform distribution.  So the answer can't be much different from the number of distinct 800-bit strings in an independent uniform random sampling of $2^{800}$ of them, which is a fraction of $$1 - (1 - 1/2^{800})^{2^{800}} \approx 1 - e^{-1}$$ of them by the usual reason.  The fact that $S$ is chosen independently at random is not relevant to this analysis; the same analysis would apply if $S = 0$.