Even with the deepest concern and the best of intentions we often approch others the wrong way.
You can't expect to use the same tactic/ manner/ approch as every one else has and get a differant result. He has already built a defence for that.
Try and discern where your cousin is at with his beliefs/how strong his conviction.
To do that your going to have to do some things the evangelical church does not promote.
First !!!! Put your bible away!!!
That is to say don't try to get him into it just yet. Don't quote from it or try to drag him to it or church.
Second!!! close your mouth and open your ears. let his talk and actions determine what direction and how far you go.Go as far as you hcan without compromizing your faith. Discern just where he's at and what he doesn't or does believe in.
This is the hard part do not pass judgment, either confirm or accept or deny his position. Just hear him out. acknowledge his beliefs or lack of them. Make some sacrifices to be with him something he will knowand appreciate. Like ,miss church a few times to spend the time with him. (The church may not approve and most likely won't appreciate it even if they understood why).
But rest assured he will and so will God.

Get to know him and let yourself be known by him. The real you.
Not the version of you that shows up in church every Sunday mornining.
Everybody has their church persona ,so don pretend you don't.

Do not compromize your christian values. But let him know that at least for now you accept him as he is.
Don't make that chritian line in the sand. You want him to be better treat him like he is or at least can be better, You want him to be a christian? treat him like he is one.
Patience, patience, patience
Open the gate for him when ever you get the chance but let him walk through when he's ready .
Once he starts confiding or moving in that direction encourage him but don't push or pull him to a commitment.
Introduce him to some of your christian freinds Get him involved through something he likes with out forcing a commitment
Casualy and slowly start talking more about your feelings tward God. Let him get him used to hearing it.
Take the time, make the effert, and if he comes to christ great if not at least you and God will know you did your best.
Oh yea and pray for him and you and that he comes to a saving knowledge of christ. and pray often.
Sincerely and
ernestly yours
and His, Cliff