Tooseevee,
I wouldn't have thought that a small battery would have the clout (sorry, Cold Cranking Amps) to power the electric starter, which is maybe why you're only getting clicking noises. The factory upped the capacity the 14AH on the E/S AVL for a reason! If your battery, hot or not, has the power to run all the lights brightly there should be more than enough for the ignition, which makes me wonder if a connection has come loose in the ignition circuit or if there's a fuse blown. If you haven't already done so check for a fuse holder with the brown wire that goes to the TCI box.
Shame it's not a small job the machine the AVL right side crankcase for a full set of timing gears and a magneto.
Meanwhile, I have had a Motobatt 14AH on my Electra-X the last two years, it has held up very nicely, but other brands are available!
Regards,
A.