There are no rules for the bridal party, although it's probably more customary for you each to have people who mean something to you, followed by people who are meaningful to the fiancé/fiancée. There's no rule about size of the bridal party, but you have to keep in mind that it's an expense for each participant. Bridal parties that are so large as to overpower the size of the regular "audience" make guests feel that they are the ones who've been left out because everyone else in the universe is in the bridal party.
I think it's very odd that your fiancé would want your third cousin - if that's what you meant, rather than his own 3rd cousin, or in any case, someone he just met - to be in the bridal party. A person you two barely know would probably react badly to being asked: "Why doesn't he have closer friends?" "How desperate does he have to be and how many people already said no before he asked me?" and so on.
It's not just people you like at first meet - it's people who are close to you, in my view. Yes, we include the siblings of our intended spouse, but it's also assumed that there has been some sort of contact in meeting those who are significant in our loved ones' lives.
Also, remember that there are other possible "jobs" or honors in a wedding that can be done by people not in a wedding party, and you can certainly take pictures with lots of people. My stepdaughter's wedding, for example, including multiple group photos of all the cousins, none of whom were in the wedding party.
What I think is more telling is that you two are already arguing and showing either a lack of understanding or a lack of maturity in how you are handling something as relatively simple as the other person's wedding party. This is kind of a test of your ability to wade through major life challenges in your lives together. Most people have to confront values-oriented issues, from religion to child-rearing to financial problems to major illnesses to caring for elderly or infirm parents. Being able to honor and manage each other's families is a huge thing. As things stand now, your fiancé doesn't see your relationship with your 2 male cousins as all that important for him to get involved in, and you don't see this 3rd cousin as a key player. Unless you can work all that out (perhaps with a neutral 3rd party in some pre-marital counseling), you have a lot of hurdles ahead in your life together.