Most US universities can't cover 100% of the costs for the students who attend there, and those that can tend to be the most elite unis in the world. You won't qualify for US student loans, as those are only for US students. What would be wise would be to apply to a couple of US unis, as well as to some unis at home. See where you get in, see what the costs end up looking like, then decide where you're going. Do not set your heart on going to uni in the US, as financially, it may not be possible.
However, the idea of doing a year abroad at a US uni as part of your UK uni degree is a good one. What I'd suggest you do is contact the US uni the year before you plan to do your exchange, and find out about their sorority system. Then contact one of their sororities before you leave home, and find out if/how you can rush/pledge while you're on campus.
Not all US unis have sororities, so for your exchange year, you'll need to research which do, and focus on those for your exchange. Note that it costs money to be part of a sorority, and you'll need to save that up so you can pay it. On average, it's about $1,500 per year without housing/food, more like $4,000 if you choose to live in the sorority house and eat their food. You'll need to earn that money so you can pay it.
You actually can work in the US while here on a student visa. The job must either be on campus, or else directly related to your field of study. There is no guarantee that you'll get such a job, or that it will pay enough; so don't consider this as your way to pay for an exchange or a sorority.