It does sound like your charging system isn't. Read the charging system guide.
One thing is you need a good fully charged battery to do the testing.
I might suggest taking your battery out of the bike and charging it up with the proper charger, one designed to be used on small batteries. A car charger has too high a charge rate.
Once charged take it to a place that sells batteries and get it load tested. If it passes the load test it will work fine for testing the charging system.
Once you get the charging system working right then get the smaller battery.
On the Shorai site they recommend the LFX14L5-BS12, 2.31 lbs, 210 cca's $159.95 can be found cheaper at other sellers.
I use the Ballistic 8 cell. It weighs 1.7 lbs, 275 cca's and $169.95. It can be found cheaper at other sellers. I got mine off Ebay, $135.
Leo