The site PM system and chat DM system are intentionally separate systems because they serve different purposes in my eyes. Though the chat DM system is currently somewhat deemphasised because the user experience sucks balls.
That being said, I do want to have some kind of notification in chat for when a site PM is received, but I'm not really sure how to implement this yet, the thing I have in mind doesn't really make sense until I've done the channel system overhaul and chat DMs are actually seen as channels internally as opposed to quirky unicast messages that exist in some limbo (thank u AJAX Chat). Even then for the Big site PM system you'd still be sent to a new tab, but at least you'd get a notification.