First, congratulations on your upcoming wedding! I’ll keep you and your fiancee in my prayers for blessings over your life together.
This is a quandary to be sure but not one that is unresolvable. It is your wedding day, you’re allowed to invite who you wish and no one has reason to take offense provided that you are consistent and fair. Oh, people take offense at stupid things at weddings - my aunt, who is also my godmother, refused to come because we didn’t give her some way of helping out (she has a way of taking over things, too). It’s not unusual to ask for adults only - my cousin last summer offered free babysitting (worked out great) at his out-of-town wedding to the rest of the family who have kids. And I don’t think it’s unfair to ask only for engaged or spouses to be brought - that keeps single adults from the pressure of having to find a date, and it also keeps that random boy or girl who is a summer fling from showing up in the family photo (ahem to the same cousin who got married last summer - we had to photoshop his wife in from our wedding a few years back).
Here’s what I would do: send an invitation to your cousin, addressed to your cousin alone. If you have single invitations (ie, that have “Yes I will attend / No, I regret I cannot come” versus "Number of attending: ____ "), send him one of these, and send all of your unmarried, unengaged cousins one of these. Adults understand that leaving off “and guest” from the address isn’t being rude, but rather a matter of practicality, especially if the bride and groom are young.
That said, for older relatives and friends, include “and guest” if they’re local. What’s a good cutoff? 20s get invited on their own, 40s get “and guest”. Between? Case by case if you need to. It can get tricky - don’t invite the summer fling who isn’t going to be around in six months but don’t make a widower or divorcee come alone.
What if your cousin brings his/her live-in gay lover? I hope they enjoy themselves and that no one should make a big deal out of it. The day is about you and your husband, and no guest should be allowed to spoil that. I mean NO GUEST. My own stepmother, my father’s wife, nearly caused a scene at our wedding because she was drunk, her mother was in the hospital 300 miles away, and she didn’t think I gave enough of an homage to my father in my speech welcoming the guests (I’m the public speaker in the family).