I would also like to get some attention on this, because as of now no Flatpak apps have access to Blackmagic hardware, not only OBS.
The problem is that the flatpak apps, being sandboxed, don't have access to the system's library files, so they try to find the DeckLink API library file that is installed with Desktop Video and they can't find it.
The outdated solution that pavola mentions has been to manually download Desktop Video, copy the library, and repackage the flatpak app with the added dependency, which as far as I understand is just a single file. All app developers could automate this, except they can't because of the way that Blackmagic makes available the download links for Desktop Video (there is not a single static link to reference). Obviously they also cannot host or distribute the files themselves.
The ideal approach would be for Blackmagic to release the library as an official Flatpak extension, that the other developers can add as a dependency and reference. That way, Blackmagic would mantain complete control on what version of the library is released and in use.
This solution would require very little work: A json file describing the extension, and uploading the library itself. It would mostly just require the interest of Blackmagic to do it.
Some additional information on the approach that OBS in particular tried to take, can be found in this link:
https://github.com/obsproject/obs-studio/issues/6004