No particularly good method yet. There’s an open PR to add an endpoint for SSH upload here, but at this moment there isn’t a good way to do it. I don’t believe native mode would help in this case. I don’t believe Kodi downloads the subs to an easily accessible location, and I’m even less sure if they let you define a location to download them too. Once the previously mentioned PR gets merged we can look into it further and try to catch the subs and upload them to the server from there.
In the meantime, I’d suggest looking into the opensubtitles plugin in the server. That might help find more