CLEP -- transfers to 4 yrs?

[studyingfortests]

You'd need to check if the selective school even allows second Bachelor's degrees. A lot of schools only allow a single Bachelor's degree, then shove you into grad school.

Interesting. A spot check of three Ivys supports your statement; none of them allow a second bachelor's.  However, most of the Univeristy of California schools, including Berkeley, do offer them for at least a handful of majors, and the University of Illinois system seems to allow and perhaps even welcome them at least at some locations.

I suppose the Ivys don't want anyone doing most of their work at another institution and then piggybacking a second degree from their prestigious programs.  But it seems like there are, at least from my tiny, flawed, and likely non-representative sample, plenty of highly respectable schools who would accept a transfer student who has already earned a bachelor's degree.  It will really come down to where the OP wants to go.