My guess is the protein is coming from the lymph vessels.
With prolonged lymphatic obstruction, you get a dilatation of the lymph vessels, which is called lymphangiectasia. This is also accompanied by surrounding fibrous growth (probably contributing to the non-pitting oedema characteristic of lymphatic obstruction).
Anyway, lymph vessels have fewer tight junctions than blood vessels, so it's likely that with lymphangiectasia, there's a greater propensity for proteinaceous fluid to escape into the interstitium, thereby increasing interstitial oncotic pressure. This increased oncotic pressure then pulls fluid out of the plasma, further exacerbating the oedema.
Milroy disease is congenital lymphangiectasia, which we likely don't need to know for the USMLE, but acquired Elephantiasis, caused by Wuchereria bancrofti, we do need to know.