Prove that context free languages are not closed under swapping prefixes and suffixes
Did you try $\{\ a^nb^nc^md^m \mid m,n \ge 1 \}$?

Or, if you insist on alphabet $\{0,1\}$,
consider $\{\ (01)^n(011)^n(0111)^m(01111)^m \mid m,n \ge 1 \}$?

And be shure to intersect with a nice regular language after the operation, but from your question I see that you know how that works.