If it were me, I would give your DC transformer a first shot, or even AC without the diodes. You already have it, and all you need is a solid state relay and timer to switch it on and off. That is quick and very cheap, if you can get a used relay.
Both methods require that the electrodes be in good contact with the wire being welded. If the contact is at all loose, the CD welder will blow the end of the wire off. I don't know whether the conventional welder is less sensitive in that regard. Also, a bank of capacitors can be quite expensive, if you buy them new. If you buy them used, and they have been sitting a long time, you will need to condition them before putting them at full voltage. That is easy to do. It is just a matter of charging them to a lower voltage and ramping it up over time.
Lastly, I also experimented with car audio capacitors to get even more capacitance for my bank. They work fine, but are more limited in voltage compared to the 35V or more that you can use with regular capacitors.
John