Okay, so here's what you do, you have to make the transition slowly over time. So for now, keep everything okay with your GF or she might cock block you. You also need to figure out if the cousin wants you.
So In the mean time you become really good friends with the cousin. Have her tag along with your and your girlfriend here and there. It'll look like you are just including her and trying to know her family better and get on good terms with them. You should also have her bring other members from time to time so it doesn't look like you are trying to target her cousin. Maybe the first time you take a member with you, you don't choose her. Don't do it too often, or she'll get suspicious. It has to seem natural. Going to a movie? See if the cousin and someone else wants to come along. Now you look like the fun guy.
Don't make it obvious that you want the cousin. She'll sense that immediately and red flags will go up.
During these outings, you need to get to know her cousin. Find something that isn't romantic that you and the cousin can do together. You want to do this in front of your girlfriend. She won't question it since it isn't like you are trying to go behind her back. This your chance to really get to know the cousin more. It's like a low pressure date, only the cousin won't realize it. She'll be much more at ease.
Overtime look for signs that she's into you. Hair Flipping, lip biting, if she applies lipstick or balm when she's around you, if she's a bit more handsy. Anything. Again, don't let anything get too far, you just need to look for queues.
If you are certain after all this the cousin wants you, you start to slowly spend a bit more time with your GF. She wants to see a movie? Damn, your busy! You really want to see it, and for some reason, the only good time to go is a time where she's busy. You don't want to go alone, so maybe her cousin wants to see it with you. Don't over do this, or she'll become suspicious. You still kind of date her, but you let things die down. You're not as spontaneous, etc. You need to make it feel as you two are drifting apart, and whatever spark there was, has gone away.
Try and still do things with her other family members though, but slowly focus on the cousin more and more. You still can't date her at this point.
Now you look for indications that your GF is losing interest. Maybe she doesn't call you as much, maybe she makes less effort in things. This is good. There needs to be a sense that you two are really together anymore, while looking like you are still in the families life.
At this point, you are informally broken up. Perfect. Now you start to spend more and more time with her cousin. Occasionally telling her cousin fun.
Now the final thing you have to do is, take your "GF" out with her cousin, and during these outings, you and the cousin have lots of fun, maybe even "forget" your GF is there. You want to do this at least two or three times. As she processes this, she'll realize that the cousin is a better match for you, and will just let you go without ever realizing this was all setup.
Next, spend more and more time with the cousin, and eventually one thing will lead to the other. Congratulations, you are now dating her cousin and no one got hurt.
You'd have to do this over a series of months though. I'd say any faster than 3 months would be too quick, but easily doable by the 6 month mark.
ONLY DO THIS IF YOU ARE SURE THIS IS WHAT YOU WANT, and YOU SURE THE COUSIN WANTS YOU.
Good luck!