Hi, I'm new here, and it seems I have a perfect timing, as I finished previous parts yesterday, just in time for the first choice of Broken Chains.

And my vote goes to:
[x] Give it freely, knowing the gift will curry a great deal of favor with both the Fujou family - and with Kaiduka and Hijiri

Avalon won't be very useful for Shinji, expect as possible bargaining chip. But even assuming that it succeed as that, Shinji can easily end up again in the way of Einzberns (or Zouken if we give it to him), and they will be stronger now.

Water will be always useful, but indecision may led to wasting too many opportunities, and personally I'm not of opinion that Shinji should ally himself with Zouken, because even if he is to regain his sanity he probably still be manipulative bastard just fighting for greater good now. And that wouldn't bode well for Shinji's forging of his own path, ignoring even possibility of him being seen as necessary sacrifice on a path to Utopia.

Giving water to Fujou's gives nice immediate advantages, and builds on already existing network of allies, allies that doesn't seem to be of the backstabbing type. And Shinji needs help now, not in unspecified future that he might not live to see.