I don’t think there’s a way to get shift-enter working in the browser, but I’ve grown accustomed to hitting Enter and then o to insert a newline. I actually prefer to use that to shift-enter. It’s still two keys. In this case it’s typing a sequence of two keys rather than typing two keys at the same time.
I stopped using the code above and switched to simply remapping enter to Esc, and wrote a quick blog post about it.