I have just had all this with my Uly.
HD say that the "state-of-charge" for a disconnected battery as - 12.7v is 100% down to 12v as 25%. Mine was 12.7v charged and still not right. Check the charge rate, with the engine running at 1000rpm or more, with ECM-Spy or a DC meter. It should be about 13.5v or more, so charging ok, but I found the neutral light was dimming so belive the battery was not taking a full charge.
Check the terminals. I would not try to start the bike to many times, if clicking, as I belive it could damage the starter. Also it could be the timing. If you have had hard starting in the past - IE A pause, on turnover, on hiting the starter button it could be that and/or a shagged battery.
For me it was a shagged battery and on the advise of
I bought a OEM unit as he rates them. £98 inc VAT
but it sorted the problem. It is a gel battery and none of them are cheap. A std acid type one is no good as the one on the Uly lies on its side.
I do not think you should fast charge a bike battery. I use 1 Amp, or less, auto control rate charger.
The Ugly has an accessory socket, doesn't it? If you can't get near the mains, I got myself one of these a few months ago, works well in the truck.
The socket is ignition switched so it will not charge through it "key out". The solar panel and socket, in use, are not waterproof.