We could just take the <A HREF="http://dlmf.nist.gov/10.19">large-$n$ asymptotic</A> of $J_n(z)\rightarrow (2\pi n)^{-1/2}(ez/2n)^n$, and then

$$f_n(\rho)\rightarrow \frac{1}{n}(\tfrac{1}{2}e\rho/n)^n.$$

This seems to be quite reasonable (blue is the exact result, gold is the asymptotic expression, plotted as a function of $n$ for fixed $\rho=10$):

<IMG SRC="https://ilorentz.org/beenakker/MO/fnrho.png"/>