with the battery being deleted the only thing powering the ignition system is the stator. the stator on bikes that are meant for not having a battery are different than those that are. think of the battery as a capacitor, storing the electrical energy. when you take that away the stator is doing everything it can just to power the ignition and will not be able to power much else. the higher the rpm, more sparks per second, bigger drain on stator and without the storage device, it cant keep up. if the bike was meant to not have a battery the stator would have alot more output for ignition and lights.
so simple answer is yes a battery should cure this problem. get a wiring diagram and make a new harness. if you dont wire battery back in properly you can start blowing things. if you need help keep posting ill help. bit of a wiring guru, wired everything from atc's to cars to heavy equipment
Trikes: All Honda: 84 250r, 85 86 and 87 250sx, 84 and 85 125m, 79 82 84 and 85 110's, 73 us 90 and 77 atc90, 75 atc70
Cycles: Honda: 71 cb100, 78 cb750 coming soon cb750 trike. Yamaha: 67 yds 3 (250cc, 2cyl, 2 stroke), 82 750 SECA
Other toys: Chenoweth VW sandrail dune buggy, old race go kart, racing mower, 76 Arctic Cat Pantera 440 snowmobile
KEEP IT ON ALL 3 OR AT LEAST 2 AT A TIME