Have we so soon forgotten the 3 CB550 'cafe' bikes, some noted here in these pages, whose Li-Ion batteries caught fire and burned them down - and one's garage with it?
I'm a lifelong Electronics and Technologies Engineer. Even I don't trust the so-called 'Li-Ion conversion' charger adapters for these older alternators: they are not just different peas from the same pod. For toys, Li-Ion batteries work fine. When pressed into actual service where conditions of use can often be outside their parameters of use, the entire charging system needs to be so designed. This doesn't start with a 50-year-old amperage dynamo of WWII design - which is what these bikes have.
There. I feel better.
The Li-Ion battery needs a charger that first senses the batteries state-of-charge and can test its existing depth-of-storage before the charging profile can be started: this must also control the regulator. Once started, the charge begins as a trickle rate that is based on the measured impedance of the battery just tested: this is why, when you drop your cellphone into its charging cradle, it flickers and turns on various icons and screens for a while before it starts charging. Then, once the lithium aligns its molecules sufficiently to reverse the charge direction to be back INTO the battery, the charging begins, then ramps up as the voltage is monitored to show that the cells are likewise aligning, until all the available cells are fully aligned (electrically) to accept the max charge rate available. This is why, when cold, it can take hours for a large Li-Ion battery to even begin to charge: when cold, Li-Ion doesn't swap electrons in its outer (atomic) orbits very easily.
Then, once a discharge rate is begun, the battery's chemistry must 'flip' direction again to supply significant current: it can do this more quickly than it can align to charge, by just losing some charge as heating. So, in a bike with a small alternator, driven around town at frequent stoplights, the battery is left to be constantly 'flipping' its chemistry as the bike discharges every time it drops below 3500 RPM (CB750 with its lights on) or 4100 RPM (CB550 with lights on) at each stoplight. This generates LOTS of heat in the battery and damages the chemistry at the same time, as each 'flip' consumes some of the lithium. Over time, this is what eats up the battery.
These bikes were designed to have a battery that can instantly switch from charge to discharge to act as a stabilizer for the whole electrical system. Li-Ion ain't it.
