There were other options...
Yes maybe Alec didn't know her last name but didn't he have other ways to track her down?
I mean most of these are probably pretty stalkerish and some are against the law but by getting her arrested was completely idiotic. Not only is she in trouble, but so is he, or did the detectives determine that she found a way into the neighbors got the stuff and then returned and in a blizzard the guy she was with can claim stupidity? Is he suppose to testify against her(doubley stupid)...
Alternatives?
He knew her first name, where and when she graduated from Pre-Med. There's public announcements about who graduates from Universities. That gives him a short list of people to search for... stalkerish? Probably.
She used his lab top. Any two bit hacker could probably figure out her passwords to the sites she logged on. A bit illegal? Probably, but probably not as bad as a felony for B&E.
I can't recall if she told him where she was from? If she did then there's another way to figure out her name by just looking at who graduated or yearbooks idk.
After GoT s7e10 credits, Lady Stoneheart kills Balon Greyjoy. It is known.