I have seen somewhere a batteryless system for the 350 twins, I just can't remember where it was. It replaced your entire charge and ignition system.
That is (was) called a total loss system.
It depends on the battery completely and then as the battery gets worn down
the engine runs worse and worse.
Racing motorcycles used that system and some did not make it through the entire race before it started to degrade to a one cylinder bike.
Keep the original electrical system unless you want to be stuck out somewhere and have to leave the bike behind. Oh...you have a cell phone? But do they have a truck?
Have you ever tried to put a motorcycle in a pick up truck?
Well if you have you will know how hard it is. You will not forget.
You probably do not like to hear that you should keep the original electrical system.
Total loss is 100% opposite of batteryless. Hence the name "batteryless"
This system simply runs a capacitor in place of the battery with the stock charging system. It's hit and miss with a 350, the charging system is pretty bad but it may or may not work.
Grab a capacitor about 15-30000 uF an about 50v (20 bucks) and put it in place of your battery and see what happens.
If you want this is work you'll need to do the following:
All led flashers
An electronic flasher relay capable of running with minimal load (ie no resistors on your LEDs)
And a headlight on off switch if you don't already have one, so that you can start the bike without the light drawing current.
Shoot me a pm if you have any question a or post up here.