Convert the last Vue instance of hand raise lead to button trigger

What does this MR do and why?

Convert the last Vue instance of hand raise lead to button trigger

• next step in the plan for scaling the use of the hand raise lead modal in &13771
• to do that we want to remove the use of the combined button/modal invocation. This is one of them, the last invoked in Vue directly.
• replace it with a button -> modal setup.
• broken off from much larger MR: !151601
• fix mobile button styling
• see #460510 (closed)

MR acceptance checklist

Please evaluate this MR against the MR acceptance checklist. It helps you analyze changes to reduce risks in quality, performance, reliability, security, and maintainability.

Screenshots or screen recordings

full screen - no change

mobile:

Before	After

How to set up and validate locally

this feature spec covers this instance of the modal launch/submission

1. Setup GDK to simulate SaaS
2. Create a new top-level group.
3. Go to admin panel and edit the group to apply the Ultimate plan.
4. Go to the Settings -> Usage Quotas page for the group and click on the GitLab Duo Pro tab

Related to #460510 (closed)