Wull...in my opinion anyone tasked with hunting terrorist would be backed by some organization that has issues with terrorists (i.e. Outcasts would care little about ridding terrorist whereas Houses would). I can't think of many vigilantes that would do that.
My best guess would be LSF, being one of the most prominent intelligence agencies, would be behind such endeavors. However, if you are going after the vigilante route...a freelancer or Merc set up would probably be best if you're looking for some sort of ignorance from Houses and pirate factions.
EDIT: As for terrorists in battleships I think support from a faction capable of running a battleship or hiring a BH or 2 would be necessary.
Id suggest Military ID or Freelancer ID, since they are FULLY independent with no affialations, and from what you say I guess you are not going to do it for money, but for your personal "faith", I dont reccomend Mercenary ID or Bounty Hunter ID, since they are supposed to work purely for money in most of the cases.
I buy things I don't want to make an impression on people I don't like.
The basic advantage Freelancer ID brings is that the bearer of it has much more potential area to sweep than any other ID bearer, as you can see a freelancer almost everywhere in Sirius.
I wouldn't go with a Military ID if you're planning on having this character flying solo (at least in RP). If you're the kind of lone ranger, the Freelancer ID is way better suited for you. Neither the Mercenary nor the BHG IDs would really say what your character is. Now, if you plan on getting anything larger than a gunship, then you'll have to circumvent your own character's story and give it a BHG ID, which would be kinda broken, to be honest, as the concept for your character isn't really a Bounty Hunter.
To aim at countering battleships will really limit your character's possibilities, though. The only terrorist battleship I'm aware of is perhaps one occasional Phantom one, but I'd say it doesn't see the light of day that often. Even Mon'Star is flying a gunboat recently. You'd end up being hunted by all other terrorists while you cross your fingers for a battleship to show up.