Wired up 5 x BCAP0310 in series now, and the server PSU also had its first task today.
I didn't mod the PSU's output voltage yet, by default it has 12.5V. Even though this is not enough yet for the capacitors (5x2.7=13.5), this setup can do 110J weld energy into 1mOhm.

I don't have balancers on the capacitors yet, but they already show that they want that.
Next will be stress testing them, in order to see if they can handle this current without damage or overheating. But I think that they will need some forced air cooling.
The PSU is doing very well, I'm impressed. With my setup below, it trickle charges the capacitors because it goes into overcurrent shutdown at ~100A every 25ms. At ~9V, it succeeds to start up, but that is partly because of that high impedance lab wiring. I definitely don't recommend doing this, because I fear that the PSU may die on that the sooner or later.
But the good news is that the PSU has a remote sense input that supposedly allows voltage control from 10.5V to 13.5V, which is spot on for the capacitor bank. The idea is: resistor limited pre-charge up to 10.5V (adjustable). Then using the remote sense input (plus an opamp plus a current shunt resistor) to charge with 70A (adjustable) current. Then keep the capacitor topped at 13.5V (adjustable).
At 70A, the PSU is theoretically able to recharge from a 50J / 1mOhm weld pulse in 0.7 seconds. That should be more than fast enough to be usable.
I'll breadboard this and make capacitor torture testing with that setup...