I assume you have checked all the cables, + / - / warning light any connection plugs, terminals all in good condition, clean and tight. You can check the wiring with a meter for volt drop with unit charging, battery + to alternator + and then battery - to alternator -, you shouldn't really get much of a reading more than .25 volt should be investigated. You can also link a cable / cables from the battery to alternator + and - to see if that makes any difference, or if you haven't got a meter or to double check on volt drop. Jump leads work but take care and don't short anything out. Get someone to help if needed.
If the warning light wire short out like mentioned earlier it could blow the diodes in the alternator, (so check asap), sometimes when the engine starts the loom moves and doesn't short out or not full short to a good earth, I have seen that before, on several occasions in fact, also the warning light wire broken in the loom, touching when engine not running then open circuit when starter throws in and when engine running. It all looks OK and can test as OK but it isn't, give the wires a bit of a tug, if it comes out of the connector or loom or streachesssssss that will be your problem, obvious area's for this are in the engine compartment, as stated above, where there is movement near the engine and around the alternator, same applies to + and - wires, give them a good tug to make sure they are firm in the connectors and not just hanging on by the crimp around the cable insulation.
The alternator is what's called self exciting, it uses the small amount of current running through the warning light and alternator rotor to create a small amount of magnetism in the alternator to start the power up. They can start up on their own if there is slight magnetism left in the rotor poles, but revs would normally need to be slightly higher, just like you have.
Wiring issues and connecting relays into that warning light circuit, like a split charge relay, heated rear window or any other devise can also cause a similar fault to what you have. I don't suspect this last thing applies to your vehicle unless you have fitted something new. I am not saying don't fit a split charge system and connect into waring light circuit, that's what I have had in all my MH's. On the very very rare occasion I have connected relays in a different way to overcome this relay problem by I doubt that is you issue, it was that or replace an alternator that wouldn't kick in unless it solely had that warning light input.
If your warning come on as NORMAL, then light goes out and doesn't glows after that, I think you probably have a wiring fault, but I could be wrong.
An ex Auto Electrician, 25 years ago, lots of things have changed. If your going down the line of replacing your alternator regulator I would see how much a new alternator on ebay was, before going down the line of getting a repair.