I've made moderate progess! I made a config file for youtube-dl to use, and manually ran it from the extension folder. I had it load a cookie file for authentication. I was able to successfully download a file from Funimation that played and everything, and get the media from Crunchyroll, but not download.
Here's the results on the command line:
C:\Program Files (x86)\SVP 4\extensions\tube\youtube_dl>python __main__.py http://www.funimation.com/shows/danganronpa-3/videos/official/despair-arc-1-hello-again-hopes-peak-high-school
[Funimation] despair-arc-1-hello-again-hopes-peak-high-school: Downloading webpage
[Funimation] despair-arc-1-hello-again-hopes-peak-high-school: Downloading m3u8 information
[Funimation] despair-arc-1-hello-again-hopes-peak-high-school: Downloading m3u8 information
[Funimation] despair-arc-1-hello-again-hopes-peak-high-school: Downloading m3u8 information
[hlsnative] Downloading m3u8 manifest
[hlsnative] Total fragments: 143
[download] Destination: Danganronpa 3 - The End of HopeâÂÂs Peak High School - 1 - Despair Arc 1 - Hello Again, HopeâÂÂs Peak High School-33861.mp4
[download] 100% of 756.30MiB in 03:40
WARNING: 33861: malformated aac bitstream. Install ffmpeg or avconv to fix this automatically.
C:\Program Files (x86)\SVP 4\extensions\tube\youtube_dl>python __main__.py http://www.crunchyroll.com/fatestay-night/episode-24-unlimited-blade-works-664495
[Crunchyroll] 664495: Downloading webpage
[Crunchyroll] 664495: Downloading media info for 360p
[Crunchyroll] 664495: Downloading media info for 480p
[Crunchyroll] 664495: Downloading media info for 720p
[Crunchyroll] 664495: Downloading media info for 1080p
[Crunchyroll] 664495: Downloading media info
[download] Destination: Fate_stay night Episode 24 – Unlimited Blade Works-664495.flv
ERROR: RTMP download detected but "rtmpdump" could not be run. Please install it.
SVPTube still doesn't detect either link though.
It looks like Funimation uses m3u8 like twitch does. (which btw, now twitch streams work in today's update, but vod links don't get detected)
So the main issue is getting it to recognize the authentication like it did when I used it manually.
I'm not sure why Livestreamer works for Crunchyroll in MPC-BE and not SVPTube, but it seems like they grab the stream as an hls stream, and not rtmp.