In theory Bob has the big picture view of the team's priorities so this might seem fine (Alice needs to keep in mind that she could be wrong and Bob may say no, Chris is working on more important things right now).
In practice this is a sign that Chris and Alice need to improve their working relationship.
If the project manager is nagging and escalating, it's because the person working on the project is not working fast enough for her. This usually happens when other priorities have taken precedence and Chris's time is otherwise allocated.
Either Chris is misunderstanding the priority of his task on the project or Alice is misunderstanding the priority of her project relative to Chris' other work. Sure, the team lead can work it out, but is that really what anyone wants? More work for Bob, Chris gets micromanaged, Alice shows lack of influence.
If Alice and Chris want to grow, they should work it out together. Chris wants to have control of prioritization so he's not micro-managed. Alice wants to get things done, but she needs to accurately assess the relative priority of her projects. And hopefully Bob has better things to do than mediate a minor priority conflict.
Alice and Chris should have lunch together and come up with a good way of working together where schedules are clearly articulated and priorities are in sync.