I first started 3 boxing and the only advantages over 2 in PvE content are that you get group XP when questing (will be significant after 60 when RAF no longer applies) and you can max out more professions. Sure, there are a handful of quests that 3 can complete but 2 might have trouble. I doubt the ability to use a summoning stone (with 3 but not with 2) is going to make a big difference.

But, like you said, you still can't run instances (unless you're way over the recommended level). For battlegrounds, I don't see why 2-boxing would be any less fun than 3. Basically BG plays out like this: If you have X toons and run up against a group of X toons, you will lose. So, whether that's 2 or 3 or 5, the story is the same. With 2, you may occasionally lose against a single well-geared opponent who gets the drop on you (hint: don't let them get the drop, play defense) but most of the time 2v1 with the ability to focus fire, you should be OK. Arena, I can't speak to because I don't do it.

If you do the numbers, you can actually run 5 boxes for just the cost of the battlechests for a couple of months with no recurring fee. (Because of the free month included in the game and the free RAF months). So, you could still give 5 boxing a try and then cancel the subscriptions before the monthlies start rolling in.

Good luck!