The published terms seem to require online booking, but permit telephone cancellation and rebooking. It sounds as if this would incur a cancellation penalty.
5X Points on Travel:
You will get one point for each dollar charged for an eligible purchase on your Platinum Card® or Business Platinum Card® from American Express. You will get 4 additional points (for a total of 5 points) for each dollar spent on eligible travel purchases made on amextravel.com. Eligible travel purchases include the following purchases made on amextravel.com: scheduled flights, prepaid hotel purchases (including hotels and flight+hotel packages), minus returns and other credits. Eligible travel purchases do NOT include non-prepaid hotel bookings, car rentals or cruise purchases, hotel group reservations or events, ticketing service, cancellation or other fees, interest charges, purchases of travelers checks, purchases or reloading of prepaid cards, or purchases of other cash equivalents. To be eligible for the 5x Membership Rewards® points, you must both reserve and charge the travel purchase with the same eligible Platinum Card® or Business Platinum Card®. To modify a reservation you must cancel and rebook your reservation. You can cancel and rebook your reservation on amextravel.com or by calling a representative of amextravel.com at 1-800-297-2977. Cancellations are subject to hotel cancellation penalty policies. If hotel reservations are made or modified directly with the hotel provider, the reservation will not be eligible for this 5X Membership Rewards® point benefit. For Business Platinum Card Members, extra points for air, prepaid hotel, and travel packages will be credited to the Membership Rewards account 10-12 weeks after final payment is made. Bonuses you may receive with your Card on other purchase categories or in connection with promotions or offers from American Express may not be combined with this benefit. Merchants are assigned codes based on what they primarily sell. A purchase will not qualify for additional points if the merchant's code is not eligible. The benefits associated with the Additional Card(s) you choose may be different than the benefits associated with your basic Card.
See membershiprewards.com/terms for Membership Rewards program terms and conditions or to learn the benefits associated with the Additional Card(s) you choose, please call the number on the back of your Card. American Express Travel Related Services Company, Inc., when acting solely as a sales agent for travel suppliers, is not responsible for the actions or inactions of such suppliers.