Use ApolloMutation component in design_management/pages/design/index.vue

What does this MR do?

Refactors to use ApolloMutation component in place of this.$apollo.mutate

Resolves #35072 (closed).

Initial work done in !19850 (closed), but in the interest of minimum viable change, this MR encapsulates changes just w.r.t. ApolloMutation

Screenshots

Does this MR meet the acceptance criteria?

Conformity

• Changelog entry
• Documentation (if required)
• Code review guidelines
• Merge request performance guidelines
• Style guides
• Database guides
• Separation of EE specific content

Availability and Testing

• Review and add/update tests for this feature/bug. Consider all test levels. See the Test Planning Process.
• Tested in all supported browsers

Security

If this MR contains changes to processing or storing of credentials or tokens, authorization and authentication methods and other items described in the security review guidelines:

• Label as security and @ mention @gitlab-com/gl-security/appsec
• The MR includes necessary changes to maintain consistency between UI, API, email, or other methods
• Security reports checked/validated by a reviewer from the AppSec team