Alternators are weird. Some of them depend on the alternator light to actually charge, if your alternator light globe is blown then everything stops. There's always current going through the bulb, for whatever reason, which might be what you're seeing.
I would disconnect the solar panel, get out the multimeter, check the battery voltage before starting, after starting and then after running for 10 minutes.
Before starting I'd expect 12.6,
After starting I'd expect 13.2 or so,
After running for 10 minutes I'd expect 13.8.
If those voltages are close then you've got nothing wrong with your system.
Edit: Another thing you can do is try turning a high load on, like a light or pump, while the alternator light is doing its flashy thing.
What I think is happening is that you have such a low load on your system, that the regulator is telling the alternator not to generate much power at all, which means that it's low enough to make the light flicker. If this is the case then everything is working perfectly.
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