Dear Cousin:
You know that I love you. You also know that I love Significant Other. With that in mind, please understand that I cannot in good conscience be in attendance at the ceremony to which you have invited me. We both know the reason why, so I'll spare you the boring details. I know you would never wish me to act in contradiction to my beliefs, and I want you to know that I appreciate your understanding.
Please know that I wish each of you all the happiness in the world.
Sincerely,
Your Cousin
"Mr. and Mrs. Smith regret that they are unable to accept Mr. Jones' and Mr. Black's kind invitation for June 1." (formal)
"Dear cousin, thank you for the invitation to your June 1 event. My wife and I are, regrettably, unable to attend. Please stay in touch! And say hi to your friend Ted for us. Love, your cousin." (casual)
Nothing more needs to be said. The ceremony isn't even given a name.
Polite people do not decline invitations with a reason they can't attend. They just decline. Those doing the inviting must accept this, and cannot presume to suspect the reason, except that perhaps a previous engagement . This is what makes etiquette work. If they press the invitee for a reason, they are overstepping the bounds of propriety. Don't fall into these traps.