If $P_s$ is small compared to $n$, then you know that Lenstra's ECM algorithm will perform well compared to the sieve algorithms, which are needed to find larger prime factors.
But when factoring a general integer $n$ of unknown provenance, the first step is usually to run ECM, just in case $n$ happens to have small prime factors. So normally this general algorithm will work fine.
The only way I can see to exploit your knowledge of the prime factors of $n$ is to carry on with the ECM step longer than you would usually do. This might lead to faster factorisation in some cases, depending on the relative sizes of $P_s$ and $n$.