Refactor design_management/pages/index.vue to use ApolloMutation component

What does this MR do?

Calls to this.$apollo.mutate have been replaced with the use of the ApolloMutation component within design_management/pages/index.vue, in line with direction we are taking with Apollo/Vue-Apollo.

Similar examples of uses of ApolloMutation can be found in design_discussion.vue, design_management/pages/design/index.vue and design_destroyer.vue.

A few tangentially-related tests have been added to ensure some pre-existing behaviour behaves the same after this refactor.

Relevant issue: #35073 (closed)

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