So I am really excited by the Bloody-Knuckled Rowdy, until I realised that the bonus feat at 2nd level
Combat Style Student
At 2nd level, a bloody-knuckled rowdy gains a bonus feat. This bonus feat must be a style feat or a feat with a style feat as a prerequisite. The bloody-knuckled rowdy treats his bloodrager levels as monk levels when qualifying for and calculating the effects of style feats and feats that have style feats as prerequisites.
is some what limited. As you do not ignore prerequisites, it seems as if 90% of style feats are unattainable. Either they have skill ranks above 2, or more than 1 feat as a prerequiste, which if you are not a human, cannot be got yet
Ideally I would love to take Dragon Style but, I need 3 ranks in Acrobatics.
What would you suggest?
I count 20 style feats that work (28%), although not all of them make sense.
I'd actually approach the issue from another direction: Combat Style Student makes you count Bloodrager levels as Monk levels for style feat chain prereqs - that meany early access Pummeling Charge. Since the alternative methods of pounce take 12 or 13 levels (Primalist and Totemic Master, respectively), that's too good to pass up. Therefore, you might want to look at style feats to switch in at the end of your turn (with Combat Style Master), like Crane Style (the attack roll penalty reduction is active even if not in the style, although the one from Crane Riposte isn't). If you don't want to spend any other feats, Snapping Turtle Style works.
I would consider dipping into Scaled Fist (taking Deflect Arrows or Dodge as the Monk bonus feat, and Pummeling Style as the bonus style feat).