Sure, it's possible. Does the alarm disable the bike, so it won't start? Some on autos do, I don't know about bike alarms.
I'd more likely suspect the melted wiring issue, perhaps melted into a ground wire. If you have a melted power wire, and that power wire is shorted to a ground wire, or a ground of any kind, then you're putting power backwards into the grounding system of the item in question, and that can cause all kinds of oddities. Direct Current(DC) voltage travels in only one direction - or it's supposed to. It flows like water, from the source to the drain. If you back up that drain, and we've all had some experience with that, then you can imagine what might happen with electrical power.
It's hard to guess at, there are a number of possibilities that could be the problem. From your description, I'm not entirely sure you'll be able to get it started without causing significantly more damage along the way. It could be OK, too - there's just no way to tell for sure without being there and going through it.
I ride an Eliminator
. What you got?