I called and pulled the trigger.

I used the number on the website (1-800-542-8680) and the agent identified themselves as being from Ritz Carlton. I did not mention any rate code, I simply said that I wanted to use my Club Level Upgrade certificate from the credit card and the agent knew what to do.

Interestingly, the rate quoted (with the Club Level upgrade, therefore I assume Z38) came out to exactly the same price as the Z34 rate I had seen online. I doubled checked to make sure that wasn't a mistake and my reservation does indeed show as a "Club Deluxe Room, Club level" with the rate details showing "The Ritz-Carlton Rewards Credit Card 100 (USD) Credit and Club Upgrade Combined Offer".

To sum it up, for my 5-night stay:
- the cheapest rate for a basic room (Member Advance Rate) was 2,532 USD
- the Club level room using the upgrade certificate (what I paid) was 2,584 USD
- the Club level room without any certificate was 3,983 USD

Seemed like a no-brainer to me. That is by far the most I've ever paid for a hotel room but I couldn't be more excited.