I would go for the following solution, using core's content translation (this is IMO the canonical way to deal with your issue):
Enable the desired languages on your Drupal installation
Create a simple translatable (custom entity type or content type) called translation, having a title and a value field (and maybe a type which refers to a contextual "...
Search API provides several general fields (e.g. Aggregated field, Rendered HTML output, URI, etc.) which are available to all Datasources.
Edit your index, when on the Fields tab, click Add fields and under the General heading add URI. You will need to re-index your content and the new fields should be available.