After a number of calls today, I finally managed to get a Chase agent on the telephone who took responsibility for answering questions about the Presidential Plus card. Call 888-795-0576 and just wait until the prompt offers you the option to speak to an agent.
Unfortunately, Chase appears to have this offer locked down extremely tight. Targeted AmEx members received a personal code. If you misplaced your code, you're out of luck. No, they can't check on eligibility for the offer. No, one cannot apply over the phone. Only option is an online application, which currently offers the $95 statement credit. Would you be interested in a bonus mile offer for the One Pass Plus card instead?
Presidents Club customer service at 800-322-2640 stated that they have no idea how Chase targets these offers or what criteria are used. And they seem to have the wrong number on file for Chase to boot.
C'est la vie. Guess I'll just have to cough up the $300 now.