If I understand correctly, the issue is that the output you select in Soundjack’s output dropdown menu is not being picked up by Blackhole.
I use Loopback, not Blackhole, to accomplish what you are doing. When I do that, I don’t worry about the output device designated in the SoundJack webpage, but instead I capture the audio output from the Chrome app.
I don’t know Blackhole, but does it have an option to capture an application’s audio output?
What about downloading the trial version of Loopback and experimenting with that, to see if you can eliminate Blackhole as a variable?