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I would start with your connections at the alternator. First and foremost, remove the main power lead, the big one and clean off the terminal (if you haven't done that already). Next, find the ground wire and do the same. If you have one going to the block--that is to the alternator mounting bracket, remove it and clean it. I once found that someone had connected the ground wire to the alternator through bolt. I think you have a power transmission problem, and once the draw increases, something can't keep up. Next step is the junction box at the left front fender behind the battery. That's the little, rectangular black box near the headlight relay. Disconnect the battery, pry the top off, pull the terminal block out of the box, disconnect and clean everything. Make sure all of the connections, even the small ones at the battery, are clean and smeared with dielectric grease. From there you need to move into the fuse box, but I'll bet it's the "high tension" side of things.
DS