I think replacing the battery is the best choice, even though it has not been through many cycles. There are many factors that can effect the lifetime of a battery. Using an incorrect charger to charge the device can shorten the battery life because the battery is likely being supplied with to high of a charge current or to low of current which can eventually cause damage to the battery and possibly the charging IC on the logic board. It's not good for a battery to leave it on a charger for extended periods of time either, because overcharging can cause damage to a battery. Letting a battery discharge to often to the point it turns itself off can also shorten a battery's lifetime. Exposing to extreme temperatures often can shorten a battery's life as well.
If the battery discharges quickly after a full charge then it should fix the issue by replacing the battery. Sometimes software may cause issues by not properly monitoring the battery's charge levels. Reference the device page to look for information about changing your battery.