Let's assume all the other parts of the system are in good mechanical condition. In a series circuit you have common current. The small battery would act as a load on the bigger battery, so the larger battery would drop voltage. The combined circuit might only deliver 18 volts at 8 amps.
I'd choose the 360 CCA 12 Volt battery. I get the reasoning behind increasing the voltage to the system. I've done this a few times with 6 volt cars in the winter time. Jump the 6V battery with a 12V battery while the engine is cranking. That way the 12V is pulled down and the car's electrical system doesn't get zapped with the full 12V. At cold temperatures there is some headroom for component heating due to excess voltage.
Another consideration is battery charging. The bike will typically put out 14-15 volts so the 24V combination is a problem. The best way to see if a larger battery will help is to jump the bike with a car battery. If this this works every time then a bigger battery is a good fix.
Most starter issues relate to how much current actually gets to the starter windings. Perfect grounds and heavy cables plus a good relay are the only ways to prevent voltage drop.
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2011 EC250E
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