A car's jumper cable output can over load a little bike battery, especially if the car is running.
NAPA sells a wet cell battery that ships with acid on the side, so it may be the case the battery is fairly fresh when you add the acid. I believe they cost around $60 if I recall. According to Consumer Reports, batteries with acid already in start to age as they sit on a store's shelf.
If you go for an AGM battery, it seems that according to some google results, they sometimes don't take to being fully discharged too well. A trickle charger is required for long term or seasonal AGM storage.
