Well, I tinkered around a bit with Speech Note which has a good amount of features and is easy to install as a Flatpak. I think it has an option to do this, but requires a bit off fiddling, an extra tool and permissions for the Flatpak. I didn’t find any software with a particularly good integration into the Desktop, though.

Also read about Blahst but didn’t try it yet. Maybe that one is an option.