Which is the fastest way to find a member of a subgroup with known size modulo prime $P$, with know factorization of $P-1$?->$x$ with $x^s \mod P = 1$
Is that the best/fastest way to go?


Well, unless you give some criteria, whether it's the best is unanswerable.  However, it might not be the fastest; you could just note that $k = f_2 \cdot f_3 \cdot … \cdot f_i$ and skip the baby-step-giant-step algorithm entirely.

In addition, if all you want is a random element of the subgroup, you don't need to find a generator.  Instead, all you need to do is select a random value $r \in [1, P-1]$, and compute $x = r^k \bmod P$ (using the above definition of $k$); if $x \ne 1$ (true with probability $1 - 1/s$), that's what you're looking for.

BTW: why do you think you need $x$ to be a prime?