Tentative parts list:
Battery- 3.7v AW IMR 18650 from my stash $0
Batter holder-
Digi-Key Part Search $2.83
Resistor- 10w 3.6ohm
SQP10AJB-3R6 Yageo | 3.6W-10-ND | DigiKey $0.63
Capacitor- 1.5F 5.5v "supercap"
Digi-Key Part Search $4.19
On/Off Switch- Not really needed IMO. I will just take my battery out when not in use.
Momentary Switch- SPST NO 2A
Digi-Key Part Search
Alligator Clips- Plenty of those in the junk drawer, might get some nice ones if this works though. $0
Fuse: 2A 125vdc
Digi-Key Part Search $1.95/qty 3
Wire, solder etc- again from the junk drawer. $0
So less than $11 before shipping for bare bones parts to give it a try. I think I will.
I calculated the capacitor charge using this calc here:
Charging a Capacitor
So at 3.7v-4.2v power with the 3.6ohm resistor and a 1.5f cap = about 5 seconds to charge to 600000uf which ought to be enough to weld the wire, if not holding the button longer will increase the charge but adding more caps may be a better solution. The resistor should limit the battery discharge to around 1 amp and the 2 amp fuse as a safety, well within the limit for the battery. 10 watts should be way more than enough for the resistor as well.
Am I missing anything here?