I’d do the year/semester abroad after being connected to the 4 year school. You’d then know which courses will transfer and get the benefit of the FA through that school. If you are a full matriculated student at the foreign school,and the school is listed on the FAFSA as a school the Dept of Ed works with, you might be able to get aid. The aid is sent to the school, they do not send checks to students directly.
A lot of schools in the US WANT you to do a year abroad. They have a lot of programs they’ve already approved. My daughter’s school has 3 ways to do a study abroad: a) their program in London which is the easiest as the courses transfer back directly and your FA can be used (including school merit aid); b) schools that the university has an agreement with, with the courses pre-approved and these schools usually give a discount in tuition; c) any other school, and the home university will try to work with you to make sure all the courses transfer but there are no guarantees and you have to arrange your own tuition, but you can use FA (although not the school’s merit aid)
Under the third plan, you might be able to do the same program you’ve already investigated, but with the 4 year school’s pre-approval and get to use your FA/Pell.