If nobody has a better idea, I will simply get a (real-variable) Taylor series for $\zeta(sigma+it)$ up to second-order with remainder. This is just (real) calculus - one can easily get the continuous continuation of $\zeta$, $\zeta'$ and $\zeta''$ up to $Re(s)=1$ by Euler-Maclaurin. Perhaps not ideal, but not horrible either.

If nobody has a better idea, I will simply get a (real-variable) Taylor series for $\zeta(\sigma+it)$ up to second-order with remainder. This is just (real) calculus - one can easily get the continuous continuation of $\zeta$, $\zeta'$ and $\zeta''$ up to $Re(s)=1$ by Euler-Maclaurin. Perhaps not ideal, but not horrible either.