Contactor rating with capacitive load, how to compute the maximum allowed capacitance?
[Battery] datasheets give me 84 milliohms, so let's assume less than 0.1 ohms for V1. For C1 ESR and cabling the addition is small, say 20 milliohms

so you got \$R_\text{series}\$=0.1 Ω in series with a short on connection, i.e. 350 V · 0.1 Ω = 3.5 kA inrush current, but for very short time (hence the importance of the parasitic inductivity)!
You can calculate the charge current curve for an RC element over time as
\begin{align}
I(t) &amp;=\frac{V_1}{R_{\text{series}}} e^{\frac{-t}{R_{\text{series}}C_1}}\\
&amp;= 3.5\,\text{kA}\cdot e^{\frac{-10t}{C_1·Ω}}.
\end{align}
Note how the value of the capacitor doesn't change the peak current, but only the duration of high current. So, basically, you're asking the wrong thing here; you don't want to find the larges capacitor for which your contactor doesn't weld, you're looking for the lowest  \$R_\text{series}\$ with which it still works!
Realistically, your cabling, contactor and connectors do have some parasitic reactance, so I'd encourage you to play around with models of these in SPICE. For a start, simply put a 10 nH in series with your battery (in addition to your  \$R_\text{series}\$) and see how that changes things in a step response.