If you just hit what is needed and don't linger it will be fine, kind of like a spot weld. it is really less heat than tig if done this way because there is heat up time.
#1.. for a light tack, the MIG with the heat turned down, an .023 wire, using standard welders mix as a shielding gas, the job will be done in about 1 second of dwell time.. and it does not require a high skill set to do.
TIG, it can be done, there will be a bit more heat, and dwell time.. but I would not let just anyone try it.. TIG, at least for me, is an art, and acquired skill, takes practice, something you would not want to practice on is a gun...
Silver solder is also an acquired skill that will take a bit more surface prep and cleaning prior to the job.. but it is the preferred method for attaching sights, etc.. Not as much heat, a bit easier on the finish..
With ANY welding process, you are going to be generating molten plasma, and it will have to be filed, sanded, drilled etc to blend it in. .. which means a refinish.. the MIG,, because, from what I understand, you are just trying to fill in and re-drill a pin hole will probably work fine. I hope you have a mill, or a large and stable drill press to poke the hole..
If you are careful enough, and it is not an heirloom, museum quality, just my shooter, cold blue may do the job. but again, with that too, cleaning and surface prep is the key to success... read the darn directions! and q-tips are your friend...
Sent from my 1986 Motorola Bag Phone.