Yes.. one option is that the engine could recharge the battery.
However, the battery must be a rechargable type. And, according to the type of rechargable, the charging rate must be appropriate for that type of battery. Otherwise the battery would be damaged or possibly become a hazard.
If the battery is a 6 volt wet-cell lead-acid type or a gel-cell (gel cells are basically a lead-acid battery which uses a gelatin instead of liquid electrolyte) the magneto alone is more than likely capable of recharging it properly.
If the battery pack is ni-cad, nickel-metal hydride, rechargable lithium ion or similar, the charging circuit will need additional components to regulate and limit current to the battery pack.. These battery types have a specific recharging schedules that must be followed.
So, is this cateye device designed to be hooked into the bike's system and to be recharged by a magneto? If it is, it probably has a lead-acid battery and if it's wired correctly, why isn't it's battery recharging?
The battery, if it's a lead-acid type, could be sulfated and will no longer accept a charge.. Or the battery could have a broken internal connection..
The 0.2 volt meter reading off the battery pack makes me think it's the latter.. a broken internal cell connection. Discharged and dead batteries generally show a voltage of about 4 or 5 volts.. I'm guessing yours is either internally shorted or broken.