I would rule that it'd heal the character's hp, as his hp does not change in the slightest from the shapechange, regardless of the monster's con, or even whether or not his own con-boosting item gets melded into the form (not sure on that).
Also, I'd say the character has a daily pool of x hp, x being his normal maximum. No matter how many spells he uses, or how many turns during any one casting he spends in Avoral form, his total usage per day cannot exceed x. I have absolutely no RAW back up for that, though. Purely for balance. (And because any other ruling, and my DM would either ban or nerf the spell before my poor character got to a high enough level to learn it. -_-)
And why can't the spell be interpretted as turning you into the same avoral each time? That's how I've always interpretted it.