just for giggles, could you turn your headlamps on before you try to crank the engine, and then see if the headlamps dim or simply die altogether when you crank the engine? If the battery can support the headlamps while cranking, that kind of blows my first guess out of the water, but it seems like a pretty straightforward current deficit. Headlamps and ignition aren't totally unrelated (it is in the same vehicle right?!?

), but barring any gremlins or a total wiring catastrophe (i.e. headlamp switch fails and somehow cooks dash wiring including ignition switch

) there shouldn't be a mission critical association between the two circuits.
The most obvious thing they have in common is that they are both consuming current, which is why I started there. If your charging circuit can't support both of them at the same time, I would expect it to do roughly what you're explaining is happening. I mean, you can drive it all day without problems? Only when you turn the lights on? strange... it would good of you to verify a healthy 14.7v when the vehicle is idling and then watch the meter while someone else turns on the headlamps... wonder if it will nosedive and then die on you? would be interesting to see....