Months of game time?! Dude, you just said you -just- started the noble origin. Which for it's entirety is only 2 (game)days... I did the noble origin and I can say for a fact that when you leave the castle (finish the origin) the dog is clean when you go into the main game.
..But if your dog being covered in blood is bothering you that much, then you are the exact reason Bioware included the option to disable persistent blood in the first place.
New details on Dragon Age: Origins 2!!
-Now with the ability to raise Show Dogs. Comb, pet, and pamper your pretty pooch and try to win a ribbon in a dog show mini game.