It's not a sticky situation what so ever.
Your a guest of your cousin, the owner (your aunt or uncle I presume or his inlaws) of the land didn't tell you couldn't shoot it which means your allowed to shoot.
Shoot and answer the questions later.
Your cousin is a selfish moron amongst other descriptive words. Some best friend you got there dude.
Honestly if I were you I would never hunt with him again.But go directly to the owner and have them make the decision.
But if I were you I would kill it mount it,and find another place to hunt.
AND find another best friend.