Well, I tested it out. I can't duplicate your problem. Instead, the first time it fires, the beam bits just don't fire at all, and only kick in the second time. This may be because I don't have surfaces on. Either way, I do know what causes this
problem. Weapon linking. Weapon linking causes some very odd things to occur the first time linked weapons fire. I don't know why. I think I may be able to work around it by instead making it a triggered force fire. I'll experiment a little.
But this is my first ship with Beam turrets, and for that matter, my second ship with turrets, linking, and a super-beam, so this is pretty new to me.
All things considered, for a first, it was fairly decent attempt in all those respects. You might want to make it so the circle bits are parented to the beams holding them instead of to the core, though, in the final product. I dunno.
EDIT: Triggered Force-Firing appears to have issues of its own, unfortunately. I'll keep messing around to see if I can get it working, but really, the beams being a bit too long the first time the weapon fires is not that big a deal.