Proof that the ratio between the logs of the product and LCM of the Fibonacci numbers converges to $\frac{\pi^2}{6}$
A full proof of this fact can be found in the paper A new Formula for $\pi$ by Yuri V. Matiyasevich and Richard K. Guy.
A brief summary: Notations I will use:
- Let $\mu$ denote the Möbius function,
- let $w_n:=\operatorname{LCM}(F_1,F_2,\dots,F_n)$.
As you noticed, we have (for large $n$) \begin{equation}\tag 1\label 1 \log(F_1\cdots F_n)\sim \frac{n^2 \ln\tau}{2}, \end{equation} where $\tau=\frac{1+\sqrt 5}{2}$ denotes the golden ratio.
So proving your statement is equivalent to proving that (for large $m$) \begin{equation}\tag 2\label 2 \ln w_m \sim 3\ln(\tau)\frac{m^2}{\pi^2}. \end{equation}
In the paper, \eqref{2} is called Chebyshev's form of the prime number theorem for Fibonacci numbers.
By rather lengthy applications of results on Arithmetic functions and the Möbius inversion formula, one gets
\begin{equation}\tag 3\label 3\begin{split} \ln w_m &= \sum_{d=1}^m \sum_{i \text{ such that } i | d} \mu\left(\frac di\right)\ln F_i \\ &= B(m)+\sum_{d=1}^m \sum_{i \text{ such that } i | d} \mu\left(\frac di\right) i \ln\tau, \end{split} \end{equation} where $0\le B(m)< 2m^\frac32$.
One can write (by elementary results on Euler's totient function $\phi$) \begin{equation}\tag 4\label 4 \sum_{d=1}^m \sum_{i \text{ such that } i | d} \mu\left(\frac di\right) i \ln\tau = \ln\tau\cdot\sum_{d=1}^m \phi(d)\sim 3\ln(\tau)\frac{m^2}{\pi^2} \end{equation} (the last similarity is a proven result from number Theory.)
So, as a result, $$\ln w_m \sim 3\ln(\tau)\frac{m^2}{\pi^2} + B(m),$$ where $0\le B(m)< 2m^\frac32$.
From this follows \eqref{2} and thus also your result.