How could I evaluate the product $$\prod_{p>2}\Big(1-\frac1{(p-1)^2}\Big)$$ over the primes, or at least get a good estimate for it? I believe it's around $0.662$.
I thought of writing it as $$\exp\Big(-\sum_{p> 2}\sum_n \frac{1}{n(p-1)^{2n}}\Big) = \exp\Big(-\sum_n\frac1n\sum_{p>2}\frac1{(p-1)^{2n}}\Big)$$ using the Taylor series for log, but I'm not sure where else to go from here.
Do it with CAS.
I may be misinterpreting what you want, but I bet you can fix it easy. I did this all on an HP prime in a few seconds. You want p to be the ith prime? If so,
$\prod _{p\geq 2}^{m|m\geq p}\left ( 1 - \frac{1}{(ithprime(p)-1)^2}\right )= M_m$
$M_1=undefined,\ M_2=\frac{3}{4},\ M_3=\frac{45}{65}.....\left (0.6601820....\right)$
Now you can check small values, my hp prime chugged for 3 seconds
@m=500, $\approx$ 0.660182008115, m=750, $\approx$ 0.660173867029.
Hope it helps,