My computer is currently in warranty (with the 235e) and I don't want to bug HP until my warranty expires next month
You're better off "bugging" them, now, while they'll still help you through their toll free phone support. After that, they'll still help you, but it will be through their online chat system. It works well enough, and they've helped me on several occasions, but it's not as fast as a direct phone conversation.
Without telling them you had already done it, you could ask them for any info they have about making the change to another of the CPU's they offer as an option for this machine.
If the 605e was bad, it seems like I couldn't even start the computer up and have it pass HP's HW diagnostics.
It's very possible. I've seen CPU failures where some small function or instruction set failed. All it takes is one bad transistor or other component, and it could have failed even after final test when it was assembled. The same is true with RAM that doesn't show any problems until you address the specific memory cell that isn't working.
It would be nice if you can test your 605e in another machine because it would support your claim when you tried to return it to the vendor. If you can't test it, and the terms of the sale allow it, just send it back. If they allow a full refund, take it, but they may not bug you as much or charge a restocking fee if you asked for an exchange.
If another chip works, you win. If not, you're no worse off than you are, now.