I don't know what the "quad's" charging system looks like, or its specifications. Or, if it makes power at idle speeds.
The Cb500/550 alternator makes about 150 watts at the alternator's peak RPM, about 5000.
The stock bike has and electrical load of about 120 Watts. So, at 5000 RPM the battery can receive charge.
The CB500/550 alternator doesn't make 150 amps at idle. The stock bike uses about 80 watts from the battery at idle since the alternator can't keep up with the electrical demand of the bike, which is about 120-130 watts (assuming lights are on). The 500/550 alternator at idle makes about 40-50 watts.
With the stock bike, the engine remains running at idle because of the battery power reserve at idle, as that is what is providing the major power demands of the stock coils (25 watts) , the lights (~75 watts), and the alternator field coil (25 watts).
A smaller battery won't endure such drain for very long. The time will depend on the actual battery rating (Stock is 12 Amp Hours.) And, when the voltage falls low enough, the coils can no longer spark the plugs, then you have a roadside ornament that only >looks< like a wicked fast race machine that will only go as fast as you can push it.
You can kick the idle up to 2500-3000 RPM, so the bike never runs on the battery after starting. This is quite acceptable for a race bike. Show bikes don't have to run at all. Street bikes usually spend time in traffic. stop light, etc. and need to idle, so they don't overheat.
You may wish to develop an electrical budget, before you ride the bike very far from the garage.
Cheers,