I think this can be answered with GNS representation theory.
Note that all is needed is that $P$ is a faithful positive map between C*-algebras.
Let $S(B)$ be the state space of $B$. Consider the representation
$$
\pi = \bigoplus\big\{ \pi_{\phi\circ P} \mid \phi\in S(B) \big\}
$$
of $A$ on some (huge) Hilbert space, where each $\pi_{\phi\circ P}$ is the GNS representation associated to the state $\phi\circ P$ on $A$.
As $P$ is faithful, it follows that so is $\pi$.
Thus $\pi$ is isometric.
Given $a\in A$ and $\delta>0$, we may therefore find some $\phi\in S(B)$ with
$$
\|a\|^2 \leq \delta+\|\pi_{\phi\circ P}(a)\|^2 = \delta+\sup\big\{ \|\pi_{\phi\circ P}(a)\xi\|^2 \mid \xi\in H_{\phi\circ P},\ \|\xi\|\leq 1 \big\}.
$$
But now each $\xi$ is approximated by an element of the form $\pi_{\phi\circ P}(c)\xi_{\phi\circ P}$, where $\xi_{\phi\circ P}$ is the cyclic vector, and this element having norm at most one means $(\phi\circ P)(c^*c)\leq 1$.
Applying this to the above supremum yields
$$
\begin{array}{ccl}
\|a\|^2 &\leq& \delta+\sup\big\{ (\phi\circ P)(c^*a^*ac) \mid c\in A,\ (\phi\circ P)(c^*c)\leq 1 \big\} \\
&\leq& \delta+\sup\big\{ P(c^*a^*ac) \mid c\in A,\ P(c^*c)\leq 1 \big\}.
\end{array}
$$
As $\delta>0$ was an arbitrary parameter, this shows the $\leq$-part in your equation. The inequality in the other direction is trivial as $P$ is positive.
