Create (by copying and removing Wikibase dependencies from) new field definitions for labels (equivalent to the labels and labels_all fields for other entity types - labels_all is used for supporting language fallback). These should be name labels_es and labels_es_all. Also these should be exposed via EntitySchemaContentHandler::getFieldsForSearchIndex.
Deployment of this needs to be coordinated with the search team as we don't know the proper procedure for adding new search field definitions.
In the PoC implementation this was done in https://gerrit.wikimedia.org/r/c/mediawiki/extensions/EntitySchema/+/1071938/1/src/LabelsField.php and https://gerrit.wikimedia.org/r/c/mediawiki/extensions/EntitySchema/+/1071938/1/src/AllLabelsField.php