Former engineeing student here. I'm not in the cycling industry, however, I think the way you'd want to get in is to go work at a company and DEVELOP A SKILL that bike companies will want- e.g. composites manufacturing, FEA, CFD. this makes you much more desirable to bike companies.
in my opinion engineering internships aren't very useful. the two I went for I really could not contribute much to the company because the level of training needed was such that I wold have needed months to get up there (CNC precision machining and Emergency Healthcare). I highly doubt bike companies will take interns ( except maybe in marketing, HR, ec kind of positions) because most of the time, they don't have the budget or the time to train a young engineer.
it would be handy too if you are especially good with your hands and can do machining, etc. I did some basic machining of my own bike parts in school. a senior went for a serotta bike building course and is just starting out his own frame building business. there is alot of money for bike companies in the commuter market as well, so if you've done projects on alternative transport, etc, it may be useful as well. I built a working prototype electric foldable trike for my 3rd year project, it was good fun. as mentioned, try for the smaller companies. if you have an impressive CV in bike-related projects they just might take you.
if you know Mark Cote ( Specialized aero engineer ) story, he was pretty lucky. MIT aero studnt with great grades, raced on his schools cycling team. they did alot of testing in their schools wind tunnel. when he graduated, the aero bug as just starting to hit the cycling world and he landed a job at specialized. but his ample preparation allowed him o make use of this piece of luck.
wish you all the best, but I think it's pretty difficult to get into the industry straight out as a graduate. Like I said, you're going to need to find a set of skills that will make them want you.