Therefore, I am looking to find a media server client that I can run that supports transcoding to this over DLNA, among other things.
Good luck with that, you wont find much, if any. DLNA is not meant to be any transcoding etc. Its a very basic simple technique.
There are very few softwares that can transcode on the serverside, and then send it over DLNA. But you mention your server is a Raspi, so please dont get your hopes up at all to do any video transcoding on that, it just doesnt handle it. And even if you find something, the Raspis ARM platform limits your software selection.
Those people that use a Raspi as mediaserver (with Plex, Jellyfin, or over basic DLNA) make sure that their media matches their clients capabilities, and as a result no transcoding is required, then even a Raspi3 can handle it.
Maybe ask /r/raspberry_pi if there is a DLNA server that can transcode and is compatible with ARM.