Migrate Code Suggestions to CloudConnector::Tokens

Broken out of #546578 (closed)

Once [FF] `cloud_connector_new_token_path` -- Activa... (#558081 - closed) is shipped, we can start enabling the new CloudConnector::Tokens factory feature by feature behind additional feature flags. This issue should do this for all Code Suggestions features that make calls into AiGateway.headers and therefore construct a Cloud Connector access token.

Currently blocked on Remove UP deny list from AvailableServicesGener... (gitlab-org/cloud-connector/gitlab-cloud-connector!199 - merged)