When scientists speak of genetic engineering, they are referring to the current procedures of using gene-splicing to obtain a desired organ from a donor species.
Does Evolution explain how we grow organs in pigs for use in humans, or does Intelligent Design explain it?
No, Intelligent Design does not explain it. Intelligent design is a theory that the universe was designed and created by an intelligent being. I'm not sure how that necessarily predicts that individuals in lab coats will be mucking around with ribonucleaic acids. Unless you propose that the Designer independently and specially created thost scientists, lab coats and all.
You are correct that genetic engineering involves intelligence and design (lower case), but in that sense so does every creative endeavor man has ever undertaken. I am not sure you want to cite Beavis and Butthead as evidence for ID - it will be hard to get it in the curriculum.