PG
While your enthusiasm is most commendable, I cannot help but think that you are using this site as a substitute for thinking about a problem.
When you come up against problems like this, walk away, have a cup of tea and let the problem go around in your head for a bit. Then come back to it and see if it make more sense.
A few tips to guide your thinking.
Q1 What happens when you charge a battery?
Which way does the current flow?
What limits the current?
What is the output voltage of the charger?
Q2 What is the voltage of the two batteries?
What voltage would you measure if you removed the lamp and substituted a voltmeter?
JimB