In JavaScript, the multiline flag changes the meaning of ^ and $:
Treat beginning and end characters (^ and $) as working over multiple lines (i.e., match the beginning or end of each line (delimited by \n or \r), not only the very beginning or end of the whole input string)
Well, with those characters out of the picture, are there any other ways to mark the start of the input string? How can I indicate that I want my multiline pattern to only match at the start of the input string?
I know I can check the index of the match using the index property of the return value of exec, but is there a way to prevent the regex engine from searching the entire string in the first place?
No, JavaScript doesn't support the absolute anchors (\A, \Z and \z) like most other flavors do. But are you sure you need them? The beginning of the string is usually the only place where you must use an anchor. Each subsequent line is automatically "anchored" by the newline preceding it.
I suggest you drop the multiline flag and make sure you explicitly consume all the newlines. I know that's kinda vague; if you were to supply a code sample and/or tell us what problem you're trying to solve, we might be able to do better.