As each $P_A$ is a product of at most $l$ elements of {$2,4,\dots,2^t$} counted with multiplicity, we have for any $\lambda>0$,
$$\sum_{A \in \mathcal{P}} P_A^\lambda\leq(2+4+\cdots+2^t)^{l\lambda}<2^{(t+1)l\lambda}<(4m^{1/r})^{l\lambda}.$$
Applying this for $\lambda:=r/l$ yields
$$\sum_{A \in \mathcal{P}} P_A^{r/l} < 4^r m.$$