The alternator conversions usually don't self excite till the rpm is up a bit. Once it excites and turns on then it should remain on under it's own power till the engine is shut down. The GM 10si alternators originally excited through the generator light warning circuit. This is what the two terminal plug was for. If jumped per alternator installation instructions, one of the pins could be used to initially get the alternator to turn on. Generally, it only had to be jumped once and then it would self excite after that.
Definitely check all battery to frame and alternator structure grounds as well as headlamp attachment. The fender supports are pretty well bolted down to the frame but crust can build up in the headlight ground pathways over time.
If that has no affect then there may be a problem in the 10si internal voltage regulator module. I would be curious what type of headlight bulbs it has, ie plain incandescent types or the halogens or what have you. Modern LEDs are expensive but they use very little current compared to incandescents, Halogens use a lot more current so that's why I wonder.
6-volt positive ground conversion kits for the 10si are available on flea-pay and generally contain all the parts that fail.