The successful and kind way to do it, let me explain what I do.
I usually give them messages in kind consideration, I hope that they answer.. If they don't answer, I look at their followers (Friends, Family) look at who they are close to. Give them a nice message, lets call him Doug. Say, Hello Doug.. can you ask (Name) if he'd ever consider changing or giving up his Instagram handle for me, reason why etc. They might ask their friend who may have thought you were anything from a spammer to a scammer (People who don't understand usernames think you can hack their whole household just by having a username) maybe message 1 or 2 close to him/her.
That's successfully worked for me quite a few times, ofcourse it's also not worked.. but it's a nice respectful way to do it. Kindness goes a long way.