I wouldn't attached the letter unless they asked for it for verification purposes. That come off too aggressive and unnecessary. If they want to verify, they can always ask and you will have the opportunity to share the letter with them.
Keep it simple and respectful. I was in similar situation and I just let the adcom know that "I am very much interested in school X and looking forward to enroll, and finance is one of the main considerations especially the other competing school is giving me more financial support. I hope adcom can consider to support me making this decision more easily". I am sure someone else here can word it better as I am at work and typing super fast. In short, do not share the letter right away as it seems to suggest that you are threatening to leave the school if they dont match the offer.