It's not clear from your message if your issue is that SODE jetways in 3rd party airports are not recognized by GSX, or they are not recognized at DD KDCA ONLY.
Since you said you ran the Live Update, which surely has fixed an issue with 3rd party airport with SODE jetways, I assume you meant they works elsewhere, but not on that airport.
Since GSX ask SODE the list of jetways from a 3rd party scenery, the only possible explanation for this, is there might be something specific about this scenery, perhaps its usage of a different menu when in SODE (Shift+D instead of TAB+S), might have a connection to the fact SODE is not sending them to GSX, as expected.