Too weak at low levels, a bit too strong at high levels, probably average or nearly so at middle levels.
At 1st-level, they give up 2 HP along with most armor and shield proficiencies, in exchange for Sneak Attack +1d6. Lame. They'll probably be 1-6 points behind on AC (depending on what Fighter you compare with; since the Battledancer won't use shields, best compare with a greatsword-wielding Fighter, in which case the AC difference will be 1-2 points initially). Their damage output will marginally increase at low levels, if at all, while their defense will suffer at least a bit for certain.
The Battledancer will probably put their 1st or 2nd best score into Dexterity to mitigate the AC loss due to lighter armor, but Strength will probably still be priority 1 anyway. Constitution priority 2 or 3, Intelligence priority 3 or 4 just to get Improved Trip and maybe, just maybe, Improved Feint, not that a Battledancer will ever both feinting to get in SA's when he could just full-attack for similar results, or flank and full-attack to get in SA's.
Prepare for mid-to-upper-level Battledancers going greatsword + WF/GWF/WS/GWS + flanking + power attack + 4 sneak attacks, each round, every round. Well, except for the first round they attack. On that round, one tumbles up to a foe, readies, another tumbles up on the other side, and then both sneak attack. With flanking, and the first attack being a trip attempt using Improved Trip, they'll eviscerate each opponent as an unstoppable deadly duo.
Then, when they occasionally face undead, constructs, elementals, or the like, they'll just have to fall back on using regular Fighter tactics without the aid of Sneak Attack, but they'll still be doing pretty well.
Their only drawback will be the combination of having slightly fewer HP (as if they won't already have a decent Constitution) along with a slightly lower AC (which of course, only serves at mid-to-upper levels to slightly reduce the amount of Power Attack adjustment the enemies use). Against many foes, none of the Battledancer's drawbacks will really come into play. They'll Sneak Attack most everything to death in seconds.
I'd lower the rate of Sneak Attack advancement, give them Shield Proficiency, and give them something like 2 or 4 bonus skill points at 1st-level specifically to be spent on Balance, Bluff, Jump, and/or Tumble. At least that's the simplest solution I can come up with.
