My wife and I are fighting for custody of my son from my Ex. If we get it, how will that affect my goal of re-enlisting?

Since the direct question is well-covered, let me offer a bit of advice, from one divorcing dad to another.
My children mean the world to me, and now that their mother is gone we have grown much closer. Custody is unresolved, but they are with me all of the time now and I expect that it will continue to be the case once things are finalized.
There is a reason kids have two parents, and they need you both. If there is ANY way to find an amicable solution where you can cooperate and co-parent with your ex, find it.

It shouldn't affect your re-enlistment at all. If your current wife is a civilian then you don't need any family care plan or anything. If you have full custody then you are good to go but if you have joint custody then you will probably have to have some type of power of attorney for your current wife to make medical decisions for your child in your absence since legally she is not the mother and has no parental rights.

You didn't give a lot of detail, so I'll provide a general response. It should effect your ability to reenlist very little. If you follow all your Service's guidelines for service members with families with regard to deployments and benefits, you shouldn't have any problems with reenlistment. Don't let the court battles adversely impact your performance as a service member whether you win or lose custody.