I'm sorry you and Junior are going through this, juniorlover.
I hadn't thought of Vitamin K supplements to help this sort of bleeding. Please let us know how it works.
Occasionally a dog will get a grass awn stuck in their nasal passage that works into the tissue and causes bleeding. Sometimes the awn can be flushed out during a rhinoscopy. If there's a possibility that this might be Junior's problem, you might want to discuss the possibility with your vet.
Most of the NSAIDs, like Rimadyl, can affect bleeding, so if he's tolerating the Rimadyl well, I wouldn't change it. Also, I think Rimadyl has about a week wash-out time before it might be safe to put him on another type of NSAID without risking gastric upset/bleeding.
Cool compresses might help stem the bleeding a bit. Also, if your air is very dry, a humidifier might soothe Junior's nasal passages.
Best wishes, juniorlover.
Let us know how it goes, and remember to appreciate every precious moment you share with Junior!