Ignition Fusible Link Blowing after Lean Burn Conversion

twood16 said
The 3 prong does tie into the choke as well but is involved in the ignition circuit using the dark blue wire that is split throughout more of the ignition system.

J2 circuit is engine run (usually blue insulated). Traditionally it powers everything critical to running the engine, which includes powering the alternator rotor (aka field).
So that's a branch of the J2.

twood16 said
The idea is that the light turns on with the key on and once started it stays on until the opposite prong of the two gives a reading to say there is proper oil pressure and thus the light goes out.

Typically if there is no pressure gage, then a lamp is used to light up when the pressure is low.
The question is why was there heat generated in the connection.
If its split as you describe, then one terminal out is to the choke - that's somewhat high current, and the other lead is to the indicator bulb, and that should be low current when its on. Be worth looking for shorts.