Consider whether an enthusiastic letter from a scholar who knows your work well might not be worth more to you than a lukewarm one from an established authority. While Dr. Firebreather may or may not be willing to pull strings on your behalf, you can be almost certain that you will inherit his enemies. Student tells Dr. Flake this is a problem, Dr. Flake balks. Student's advisor and Dr. Flake are buddies, so the student goes to the other members of committee, who tell Dr. Flake to adhere to dept regs. Real example from last 3 years.
Student calls outside member, who is tenured, high profile faculty.
Outside member calls chairman of department and informs chair which grad regs have been broken, and that student has grounds for lawsuit, better for dept and university if dept reconsiders handling of situation. 2 days later student gets call from dept. Make sure the outside member is familiar with your home department's qual exam procedure and policy. This ensures no mis- understandings and leaves a paper trail in case of later trouble. I also include grant application deadlines, so they will know a couple months ahead of time when I will need letters.