Choosing Between the Double Diamond and Design Thinking for Two Projects

You're advising two teams:

• Team A is doing a one-time, three-month engagement to define and ship a brand-new checkout flow for an e-commerce client, with a clear start and end date.
• Team B owns an existing, live product feature indefinitely, iterating on it every sprint based on ongoing usage data and support feedback, with no defined "end" to the work.

1. Which process framing — Double Diamond or Design Thinking — fits each team better, and why?
2. What would go wrong, concretely, if each team used the framing suited to the other team's situation instead?