FYI, @pedronis and I discussed this in Vancouver. It is unfortunate that while does not need classic to install since it can use system-files to install itself into the target browser, it will end up running outside of confinement since the plugin executes within the browser context. That said, browsers like firefox and chromium internally run extensions with different privilege levels than the rest of the browser.
Looking at the browser snap case (eg, firefox, chromium, etc) these browser snaps could expose their native-messaging-hosts directory via the content interface, for snaps like jabref to connect and place files. While this plugin mechanism is not currently implemented, it is well understood.
For now, we are prepared to grant jabref use of the system-files interface in the prescribed manner with the understanding that this is the equivalent of classic. @LyzardKing, I suggest working with mozilla for firefox and @oSoMon for the chromium snap on how they might provide native-messaging-hosts via the content interface for your snap to plugs.