Can you use the X-men TA with wolverine at the begining of your turn then use wolverines trait to heal him of the TAs damage, similar to how you can outwit defensive powers to make peoples take poison damage? That would make him a very useful battery for an xmen team if you could.

No, because once you take a non-free action, it is no longer the beginning of the turn.

Unfortunately, no.

The beginning of the turn ends when a figure is given a non-free action that doesn't specify that it is or can be done at the beginning of the turn. This is clarified in the player's guide.

Since X-Men TA healing is not a free action, and does not specify that it can be used at the beginning of the turn, all "beginning of the turn" things must take place before it occurs.

Mind you, he's still an incredibly useful healing battery, he just spends more time on his second click when he does it.