The voltage with the engine running will vary based on the charge level of the battery, the temperature, engine rpm, etc - Anything from 13v to 15v+ could be considered "normal" depending on the conditions.
The charging light simply shows whether the battery is charging - light off, or discharging - light on. Since the voltage on fully charged flooded-type 12v battery is around 12.6 v (depending on temperature) any voltage above 12.8 or thereabouts will be "charging" the battery and the light should be off.
I would guess that if no other change was made, it's the LED itself that is not playing well in the system. Frankly, I'd just put the incandescent bulb back in. It's not like there's any useful reduction in electrical power by using an LED there. With the engine running and the charging system operating correctly, the light would not be expected to illuminate at all except maybe at idle if the idle speed is set rather low.