Sorry, to make this work well, I'd have to bump and commit the version number every time, and then somebody on the brew cask team would have to accept a pull request every single time. It's just too much work for nightly builds.
You can look into the portable package though. That one contains an auto-update script that can be used to quickly upgrade to minor revision bumps. You can either use that or modify the script to make it update the jar in the macOS application folder.
I see.
I believe the file can be used ad-hoc by whoever wants to try it (manually changing the versions).
Reading through homebrew docs it seems they support a :latest option for version which would replace the 4.8.6 with a 'latest' string. This, paired with a dedicated appcast for beta might be a workaround.
I only work in black and sometimes very, very dark grey. (Batman)
While I don't plan on officially adding a filebot-beta cask, you can quite easily create your own cask descriptor and maintain it on behalf of the community:
Since you won't be able to use SHA-256, I recommend using the GPG public key / signature options to verify the integrity of the download (not sure if brew cask fully implements that at this point yet though, the options are there, but not doing anything last time I checked).