We're not looking at how the French or nobility would construe the name, as this was an account 350 years (13 generations) after the fact and not by the royal court. In looking at city and court Calendar and Charter Rolls from that century regarding "Roberto de lo Hode' (de Linton)", aka "Robin Hood", there was no continuity with regard to naming conventions, (ie, "da", "de la", "de le"; "Hud", "Hod", "Hod'e"; etc.). It is not expected that grammar at that time would be specific or accurate.
While although "la" may be in question as you suggest, Captain John Rush was deliberate in his capitalization of "Roix". And, the name change was from "Roix" to "Rush" as witnessed by countless generations of Rushes, not as in "Croix" to "Cross". It is doubtful "Roix" was ever in error.