Chore: Integrate new api client into StorageFileBrowser script

The StorageFileBrowser script, https://github.com/withstudiocms/studiocms/blob/main/packages/studiocms/src/virtuals/scripts/StorageFileBrowser.ts currently uses raw fetch requests to the new Effect HttpApi based API that has a documented and typed API Spec and clients. This should be updated to utilize the new `integrationsClient` from https://github.com/withstudiocms/studiocms/blob/main/packages/studiocms/src/client/apiClient.ts#L37