Limit of the ratio of the square root of a Mersenne number to the product of its prime factors

Question

Mersenne numbers with prime exponents are numbers of the form $M_p = 2^p-1$, where $p$ is prime. Suppose that $p$ is such that $M_p$ has exactly two prime factors, $\rho, P$. Given $\epsilon > 0$, can we show $$ \dfrac{\sqrt{M_p}}{\rho P} < \epsilon $$ for $p$ large enough with $\omega(M_p) = 2$. Of course if no such $p$ exists, the result would be vacuously satisfied anyway. Supposing that there are large enough such $p$, can we show, say that the non-square-free part of $M_p$ is small enough to imply the result?