Custom spot welder design for very small scale
I have slightly grave doubts about the welding to copper bit (what are the electrodes going to be made out of so they don't weld to the wire...), but on the electronics question, there was a spot welding project in the magazine Digital Machinist. It used a car battery and MOSFET or paralleled MOSFETs to control the current. I think the first of the multi-part installments was in Spring 2017. 

Traditionally spot welders switch the primary of a step-down transformer, either to control mains power into it or to discharge a capacitor bank into the primary. Thyristors (or before that, a small thyratron) are used. The switched current is less by the ratio of the transformer, so a modest primary current can control many kA at the electrodes. In the case of the mains power, you can control the voltage with a variac and switch some number of cycles. In the case of the capacitor discharge, you have the capacitor voltage to play with, and potentially you can switch the discharge off if you use a suitable semiconductor (ie. not a thyristor).

Rather than fiddling around on your own, however, I would suggest talking to the engineers at a first class manufacturer of high-end precision spot welders and seeing what they have to say. No doubt they've run into something like your application and can advise you, maybe even do some test welds if you are planning on buying their product. Precision spot welding of relatively exotic metals is not uncommon in the manufacturing of instruments.