The current level is very good, but as the capacitance is only 3F, this means that the voltage will drop very quickly, and the power that they can deliver into a weld is limited. You wrote that you are charging them to 10V. When they get discharged to 5V during a pulse (current will be only half at that moment), then the delivered energy is 0.5 * 3F * (10V^2 - 5V^2) = 112.5J. With an assumed system efficiency of 15%, this means that you should be able to reliably weld with up to 17J (kWeld setting). This is just enough for 0.1mm pure nickel, but thicker material needs bigger capacitors.
It is assumed that with 100J I could weld nickel of 0.3mm and this power is the one I need to weld nickel of 0.15. It seems that the capacitors deliver enough power (1370A), or not?
Your current capacitors have too little capacitance to hold up voltage during the pulse. They drain too quickly, and the current drops linearily with the voltage of course. Power into a resistive load (the weld spot) is proportional to current to the power of two, so that drops even quicker. So the answer is no, it is not enough for 0.15mm nickel.
At the moment I do not intend to weld thicker nickel and the solderings to 100J in nickel of 0.15mm serve me.
For 0.15mm nickel, the required energy should be 20-30J (estimated, I don't have that available), not 100J.
The problem is that if I add Kweld in my welding head and restart it to each solder, then the power will be mismatched again, and I would have to reconfigure the soldering iron after each solder! That's why I've thought about adding a small independent source of energy.
As I described, you can add an aux power supply if you like. But I would not recommend to discharge the caps to such a low voltage during a pulse, as described above and in my previous posts. Limit yourself to 17J and see what nickel thickness you can do with that. If you go higher, the results will not be as repeatable as you would expect, which is because the pulses get too long, current gets too low, and heat gets spreaded around the spot too much.
Can you think of a better solution?
Add more capacitance. I gave you a formula in my last post that allows you to calculate how much you need based on the pulse energy that you want.
When will the supercapacitors be ready?
I will receive the PCBs for the first prototype tomorrow, and the further timing depends on the results that I get.
I've made a quick simulation to visualize how quickly the output power from your capacitors drops. You can see that after just 4 Milliseconds, you are down from 13.6kW to 4.1kW: