This is just my take on it since I have an 08 and have not suffered these issues. Sorry it's a bit long winded, but you asked, so I'll give my $.02 input.
Dead battery, bike dies, the most likely culprit is the stator being fried or possibly a bad VR and relay if it's got the wire harness "upgrade/fix". A new battery is probably in order, regardless of the outcome.
High voltage, bad regulator. Check the voltage output from the stator for sure though.
Low voltage, bike won't stay charged, stator, possibly the rotor magnets are loose/broken but not very likely.
If the battery is weak to start with, give it a full charge and see what happens with it, if it won't take one or keep it, replace the battery. +1 to the AngryGnome on the connections, make sure everything that can be tightened is tight.
There's a pretty long thread/section over on badweb that covers various problems/fixes. My personal opinion, it's a heat related problem and a crappy VR. Buell's aren't the only bikes to suffer from charging problems. I'd go with the EBR wire edm hole fix if you can, look into the charging system relay (I'd need to find it on the print, I haven't looked for it on the bike either) and possibly a better VR.
One last thing (keep in mind I'm NOT an EE) that the "fix" from HD with the wire harness "upgrade" was to me, a total BS patch that had no chance of really fixing the problem. As the engine turns over, the rotor is spinning. That means the stator is generating AC electricity ALL THE TIME. It may not be much, but it's cranking out something. At 3000 rpm, you should see 45-55 volts AC from all of the 3 yellow wires coming from the stator, voltage should be very close between them. I think (could be wrong here) HD put in the harness with a relay to cut out one leg. Makes absolutely no sense to me, but that's just me and my opinion.
What all that tells me is, at ANYWHERE above 3k rpm, it should be charging the battery, so the claims made about needing to ride it above 5k rpm to charge are BS as well. Most likely, the break point for the charging system is even lower, ie; 2500 rpm or so where at idle, it won't charge the system, but above 3k rpm it will. 2500 rpm might be the break point where it's just maintaining what it has, but not putting enough back in to recharge the battery.
Again, take all that with a grain of salt/pepper/chili or what have you, I'm just a hack mechanic. I do NOT have a degree in anything. If it were me and I had to fix it to last, I'd rewind the stator to spec for an 08, get a high quality VR and remove the wire harness upgrade from hd if it was installed. I'd also SERIOUSLY consider getting the EBR rotor kit. Yeah, it's a bit of coin to get one, but since it looks like it's a heat related problem to me, I'd go that route anyway and I'm not made of $$ just like everyone else.