I know this is a really late reply, but I believe I have found a way. The problem is that it's very manual work-ey, but so far it works for me:
First, you need 2 things: uBlock Origin and JDownloader 2.
I'm not sure if this is totally necessary, but add your account on JDownloader 2 under play.dlsite.com just in case. Keep JD2 open.
Anyway, open up the logger for uBO, then on your DLSite Play, navigate and open the video file you want to download.
Now, check your Logger. There should be an entry with the url: https//play.dlsite.com/api/video/playlist?token=
Copy the whole url (in my case the url ends with .m3u8. Don't worry, just do it). If everything goes well, JD2 will capture the link and in LinkGrabber it'll show a video file with the name playlist...and in my case it's a .mp4 video even though the source file is supposed to be .avi (I think DLSite streams file in .mp4 to save bandwidth, since most .avi files distribute by groups are uncompressed, while .mp4 in most case are compressed), but that doesn't matter. If that appears, then congrats. You're nearly done!
Download the file immediately, since the URL has expiration time tagged in. I think if the link expires you'll have to redo the whole thing again.
Congrats, you just downloaded the web exclusive file! If you're not sure, also play the downloaded file side by side with the web one to verify.
I haven't tried to grab the other file types yet like .mp3, .wav, or various picture files, but the principle of it should be the same...I just need to find the proper link in the Logger, especially since those won't be under /video/ due to being of different file types.
Update: For audio files, you'll only get .mp3 as an output, no .wav, sorry (again, I guess it's bandwidth). For image files...ugh, it'll be a jumbled mess and you'll have to do some jigsaw solving using image editors to get it back to its proper shape...but it might work.
Anyway, in those cases, the files to download are under play.dl.dlsite.com/content/work/doujin/(work ID)/(work ID)/optimized/ , at least for my case.