You are correct in that turning the lights on or off also adjusts the alternator output.
The T350 does not have a voltage regulator. It has a just adequate alternator & uses the battery as the voltage regulator.
Your bike originally had a rather dismal 25/35W headlight.
A 35/35W H4 is ok & much brighter. But the commonly used 55/60W H4 is a big increase in load.
The wiring diagram for the 1977 GT250B is here:
download/file.php?id=826
The wiring diagram for your bike is similar, except that the GT250B has a voltage regulator and your T350 does not have one.
Your ignition switch has 3 positions:
1) off
2) on
3) parking, which has the bike switched off & just powers the taillight
In the on position, lights are controlled by the lights on/off switch on the left handlebar.
Take a look at that switch in the wiring diagram.
It turns the lights on/off (Gr & O wires)
It also turns on/off part of the alternator (R/G & G/W wires)
The alternator actually has 6 coils. The coils that are always on are referred to as the charging coils. The coils that are switched on/off with the headlight are referred to as the lighting coil.
It does seem fairly common that the wiring for the lighting coils get messed up.
Look at how the alternator is connected.
You might have a problem with those connectors &/or the lights on/off switch.
If your bike charges the battery when the lights are off then the rectifier is ok.
If this is a recent problem & your bike used to charge the battery with the lights on, then the alternator is connected correctly.
What is the history of your bike? Is this a new problem?
Its likely NOT a problem with the alternator, but you should check it.
The alternator specs are on page 15 of:
http://www.ozebook.com/compendium/suzi/suzidata1.htm
Check the alternator wire resistances:
1) between G/W & Y/G wires
2) between G & Y/G wires
3) between R/G & Y/G wires
4) between each of those wires & ground (a spot on the motor that isn't clear coated.)
You should get a small reading for test 1 thru 3 & open circuit for test 4.
Your problem may be solved by cleaning the alternator wiring connection & cleaning the light on/off switch.
You have a choice in ways to fix this.
An option is to replace the rectifier with a modern rectifier/regulator & wire it up for maximum alternator output.
This will provide a bit more alternator output & allow you to switch to an AGM battery.