Lol I can give up the ritual
Wouldn't it still be Claw/Claw/Bite or claw/claw/tail at 1 and Claw/Claw/bite/ claw/claw/tail because you can only take 1 bonus action?
The armor for flight doesn't matter character is forbidden from using it.
Path of the beast Barbarian only gets one of these abilities that they choose each time you rage. So you get either tail, or bite, or Claws. You'll get three claw attacks at 5th level if you pick claws for that rage.
There are options for dragonborn to get flight at level 5 by virtue of their race.
I would suggest that you roleplay your restrictions rather than try to treat them as flaws to gain extra power. No DM worth their dice is going to agree to a requirement that you kill another player's character just because that player didn't want to see you lose your character (which you are obviously very invested in) when it was bleeding out.
Letting you get all your level 3 abilities at level 1 is a big bonus in itself. I would suggest not trying to get more abilities from several different subclasses as part of your progression unless the other characters in the game get similarly powerful gestalt-type houserules.
For your first feat, I'd suggest asking if you can take the dwarven racial feat that lets you spend a hit dice when taking the dodge action. That will get you some method of in-combat healing.
How does your group generate ability scores? Roll, point buy? What other characters are in the party, or what sort of gaming group is it?