Hi,
I have to agree that the reason you may be measuring 11v instead of 12v is
because it is a full wave rectified sine wave with no filter capacitor.
With a load of 10 megohms and no cap the output could measure between
10 and 11 volts, roughly, depending on the exact transformer turns ratio
(sometimes set a little higher than 12vac) and the drop of the diodes.
To make sure this is really the case, add a small capacitor, even 0.1uf,
across the output and measure again with the same meter. You could
see a dc voltage much higher than 12v, like 16vdc. The cap should be
rated for 20v or higher.
If the result of this test shows 16v (or near that) then the device is
most likely working just fine. If the result shows 12v or less, then the
device is not working properly. If the result shows a voltage between
12v and 14v then the device may or may not charge the battery in
a reasonable amount of time, you'll have to try that.