I've been using sway for a few weeks as well. I just used it once to share a file from my phone, and here's what I did:

    export $(dbus-launch)
    kdeconnect-cli -l
    *then I sent the file from the phone*

It could be launched on start; there's also appindicator-kdeconnect that should work with a tray icon, as well, AFAIK.

It's supposedly UI agnostic and just requires QT5. Probably you can build it from source, and it will open as a normal app, in its own tile.