13.5V is about what would be expected from a typical single-stage converter. That much seems reasonable and indicates the converter is working.
11.2V is pretty nearly dead for the battery. Since it's been sitting and charging for a good while (a day at least at 13.5V), it should be somewhere in the rough vicinity of 12.5V at least, and usually more when first disconnected from the charger due to having a surface charge. 11.2V would suggest that it is a faulty battery, possibly with a shorted cell, and I'd have no hesitation with bringing it back to the store you bought it from for an exchange...and rather insisting that it be exchanged as faulty, even if there's someone there insisting that it's fine because some mystery box says it is okay.
The four-light battery test thingies are not too useful due to their inherent lack of precision. It would be a little like having a speedometer that only indicated stopped, slow, medium, or fast for speeds.