IPad screen broken, cannot unlock – need to backup

Unfortunately not. According to this knowledgebase article, Apple has it set up to only automatically back up when it's connected to Wi-Fi, connected to a power source and (unfortunately for you) the screen is locked.

You can, however, get in the habit of doing a backup before you go to sleep, kind of like setting your alarm clock. Just a suggestion, but I think this is the only way to make it happen daily without jailbreaking (truth be told, I don't know of a way to do this even if you did jailbreak).

Good luck.

No.

If you could simply override the setting from the computer, that would defeat the purpose of iOS not exposing internal data until it has been approved through the touch interface.