First, the bound that you cite has been significantly improved by Alpöge and Ho. Here is Theorem 1.1 therein. Let $A,B\in\mathbb{Z}$ satisfy $\Delta_{A,B}:=-16(4A^3+16B^2)\neq 0$. If $\mathscr{E}_{A,B}$ is the affine integral model $y^2 = x^3+Ax+B$ of the associated elliptic curve $E_{A,B}$ over $\mathbb{Q}$, then the number of solutions $(x,y)\in\mathbb{Z}^2$ to $y^2 = x^3+Ax+B$ is $$O\Big(2^{\mathrm{rank}(E_{A,B})}\prod_{p^2\mid \Delta_{A,B}}\min\Big(4\Big\lfloor\frac{v_p(\Delta_{A,B})}{2}\Big\rfloor+1,7^{2^7}\Big)\Big),$$ where $v_p(n)$ is the greatest nonnegative integer such that $p^{v_p(n)}\mid n$. Considering the fact that the number of primes dividing $n$ has maximal order $O(\log n)$$O((\log n)/\log\log n)$ and normal order $O(\log\log n)$, this bound considerably improves upon the one you mentioned. Keep in mind that many people now believe that there exists an absolute constant $c>0$ such that $\mathrm{rank}(E_{A,B})<c$, so the rank contribution to this bound is widely believed to be negligible. If you don't like the big oh notation, see Theorem 1.2 therein for a completely explicit version.
Second, by "Bhargava's seminal paper," I think you really mean "the seminal paper by Bhargava, Shankar, Taniguchi, Thorne, Tsimerman, and Zhao." Don't forget the other authors, even if they are numerous!
ADDED: I should also mention the Helfgott-Venkatesh bound
$$e^{O(\omega(\Delta_{A,B}))}1.33^{\mathrm{rank}(E_{A,B})}(\log|\Delta_{A,B}|)^2.$$
Here, $\omega(n)$ is the number of distinct prime factors of $n$. This might be stronger than Alpöge-Ho, depending on the prime factorization of $\Delta_{A,B}$. We can now take their minimum:
$$\ll\min\Big\{2^{\mathrm{rank}(E_{A,B})}\prod_{p^2\mid \Delta_{A,B}}\min\Big(4\Big\lfloor\frac{v_p(\Delta_{A,B})}{2}\Big\rfloor+1,7^{2^7}\Big),e^{O(\omega(\Delta_{A,B}))}1.33^{\mathrm{rank}(E_{A,B})}(\log|\Delta_{A,B}|)^2\Big\}.$$
