Parts

Relay

Schematic - Push Button Fire

The Build

I ended up using a Sonoff SV since it had everything I needed. The Sonoff SV also has 3 GPIOs available which I used for a RGB LED. It also exposes the GPIO-0 (button) which I used for the foot pedal. The foot pedal made things so much easier.

I originally powered the SONOFF SV off of the battery but found that it would cause the SONOFF to reboot sometimes. So, I ended up powered it with a dedicated 12V 2A power supply.

I programmed the SONOFF micro-controller (ESP8266) to open the relay for a number of microseconds in order to perform the weld.

The time required to weld really depends on your battery source. I noticed that when the battery was running low, it took more time to get a good weld.

I programmed the micro-controller to perform the following tasks:

• welding would be performed using 1 of 4 durations (25, 50, 75, 100ms)
• pressing the foot pedal quickly would send 12v to the solenoid for the currently set duration.
• pressing the foot pedal for at least 2s would switch to the next duration. 
• each duration had an associated color (LED):
  • blue - 25ms
  • green - 50ms
  • orange - 75ms
  • red - 100ms

Revised Build 

I revised the build to allow the weld time to be configurable via a web ui running on the microcontroller.

Observations

With my batter fully charged, a 30 ms weld did a good job most of the time.