Swap out ref-switcher for badges partial

What does this MR do and why?

Swaps out the branch selectors from HAML templates to the vue component version.

Describe in detail what your merge request does and why.

Screenshots or screen recordings

In Settings -> CI/CD -> General Pipelines -> Branch selection dropdowns.

Before	After

How to set up and validate locally

• In a existing project navigate to Settings -> CI/CD -> General Pipelines -> Branch selection dropdowns.
• Clicking a branch should reload the page with the recently selected branch. (Same behavior as before)

Numbered steps to set up and validate the change are strongly suggested.

MR acceptance checklist

This checklist encourages us to confirm any changes have been analyzed to reduce risks in quality, performance, reliability, security, and maintainability.

• I have evaluated the MR acceptance checklist for this MR.

Related to #369963 (closed)