Let $E/\mathbb{Q}$ be an elliptic curve. An old formulation of the BSD conjecture is that $$\prod_{p \leq X}\frac{|E(\mathbb{F}_p)|}{p} \sim C(\log X)^r$$ where $r$ is the rank of $E/\mathbb{Q}$, and $C$ is some constant depending only on $E$.
My question is: given $E$, is it possible to estimate $C$ quickly/ without just essentially computing $r$ first?
I don't think you can access $C$ without knowing $r$ first, and that asymptotic estimate is going to approached quite slowly and erratically, assuming it is true in the first place.
And the truth of that asymptotic estimate (for some $C$ and $r$) lies much deeper than the usual BSD, since it implies GRH for the $L$-function of $E$. That fact, and a formula for $C$ in terms of the first nonvanishing term in the $L$-function for $E$ at $s = 1$, is due to Goldfeld. See Theorem 1.1 here, which has a generalization to other partial Euler products.
By the way, at primes $p$ of bad reduction, replace $|E(\mathbf F_p)|$ with the number of nonsingular $\mathbf F_p$-rational points.