I presume you want $\mathcal{O}$ to have characteristic zero and $F$ to have characteristic $p$.
Consider the case where $p=2$, $G=C_2$ and $P=G$. If $\mathcal{O}$ is the trivial $\mathcal{O}G$-module, then there are module homomorphisms $\mathcal{O}\to\mathcal{O}G\to\mathcal{O}$ composing to multiplication by $2$.
If there were a functorial Brauer construction defined over $\mathcal{O}$ then this would induce maps $\mathcal{O}(P)\to\mathcal{O}G(P)\to\mathcal{O}(P)$ composing to multiplication by $2$. But $\mathcal{O}(P)$ would be $\mathcal{O}$ and $\mathcal{O}G(P)$ would be $0$, so this is impossible.