Will this capacitive discharge welder design work?
As shown with just a schematic, No this is not enough.


There is a bit of Physics involved in the fusion of dissimilar metals using a 3rd electrode metal (titanium?) on the surfaces that will not melt during the welded of the wire. 


  Let me try to explain this without a lot of theory and formulae using analogy of a small fuse ( and why your task is more challenging)


We know the properties of a fuse from datasheets. The time to heat up the thin wire to liquid then  separate is non-linear with A-s and A²-s of the thermal and electrical effects.  The electrical resistance is expected to rise linearly with temperature as the power increases by I² until it opens.

The fuse always has a Positive Temperature Coefficient (PTC) for R vs Temp.

Fusing a copper wire to a stainless steel battery electrode wire surface also has an initial resistance from surface area and pressure but the resistance drops when the metals fuse together so you have PTC medium of metal with (hopefully) a lower resistance when completed.  So you have a negative incremental resistance over a short time interval. So it behaves like an NTC resistance yet somewhat abrupt like an arc ( which also has an abrupt negative incremental resistance) but over a much smaller dynamic range of R before and after. 

Designing the resistances to make this a constant R is tricky as the heat dissipated is always proportional to resistance Pd=I²R thru each metal.

This means your electrode must have very low resistance contact and also a very high melting temperature like Tungsten to clamp the copper to the stainless-steel so it does not fuse to the surface so it does not stick to the surface after the pulse current is discharged. 

You also have to consider the resistance of your coils to be less than the fused resistance of your wire-bond for efficiency. This means much thicker coil wires than your wirebond.

What if we apply the fuse characteristics of I vs t but reduce the energy level so that the interface bonds instead of fusing open circuit?


  However, the fusing formula is not a simple function of current or energy (A²s)  or constant Amp-seconds. 
  



But, what you want is the fuse the interface of the wires and SS electrode and be mechanically strong.


This is why I said it is complicated to find the right electrical energy for diffusion bonding thermal energy for an electrical time interval with an impulse of some shape for a piece of copper wire to SS foil battery terminal..

So what will work?

( from others on the web)


Buy a spot welder or DIY with the power of a car battery instead a cap bank.


make a one shot flipflip with relays, one being a 1kA car starter solenoid

adjust the contact pressure to control the inital  interface surface resistance which is essential to start the heat rise process. 


fusion interface must be higher R than electrode and battery ESR resistance)


 ref