Automatic iCloud backup without locking iPad
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.