[backport v2.10.2] Badge Background Color is not applied successfully from Cluster Dashboard page

[github-actions]

This is a backport issue for #12582, automatically created via GitHub Actions workflow initiated by @gaktive

Original issue body:

Setup

• Rancher version: v2.9.3, v2.10.0-alpha11
• Rancher UI Extensions:
• Browser type & version: Firefox 131.0.2

Describe the bug

Using the paintbrush to edit the Cluster Appearance --> Badge Background Color from within the cluster exploration view does not successfully apply the color change.

To Reproduce

1. Navigate to local cluster explore page by going to Cluster Management --> "Explore" button on "local" cluster
2. Click the paintbrush next to "Cluster Tools" towards the top right of the page
3. Select the checkbox for "Badge Background Color" and choose any color
4. Click Apply

Result
The color is not applied.

Expected Result

The color should be applied.

Screenshots

Additional context

Note that the color in the "Badge Background Color" selection does successfully change, it just does not apply it to the badge itself in the top left.

[izaac]

@a-arias this is ready for test.

[a-arias]

TESTS

Rancher : v2.10.2-alpha2
UI: v2.10.2-alpha4

Fix validated ✅