I haven't fought a battleship lately, and don't think I've ever flown Nightmares or Krakens. But it is possible they could have shot it down. Which has the stronger shield/armor? I did find that's the one you send up to hit it first, let it start shooting at that one, then the others move in. But I don't think those in combination were really enough to take a battleship unless you're really good, and really lucky, so probably still a good call.
Mercenary base . . . yeah, without awesome weapons I wouldn't bother. Again you have to be really good, and really lucky. I did get a bit lucky in one game against the Academy, when someone with a rocket launcher went berzerk, killed herself, and two other academy troops, both with pretty heavy weapons. And getting a shot at a Provost using an Electric Lasso, the main reason I even went in the first place!
But, if you are equipped and good enough to handle Mercenaries, you can get very good stuff there. They are among the best equipped, and unless you'd already broken their whole line their higher-ups open up good stuff too.