I think your question is a complicated one. The four regiments of the Vistula Legion had adopted French dress by 1812 and would have had white shako-cords for all rank and file, silver for officers. Grenadiers, fusiliers and voltigeurs were distinguished by red pompoms for the first, white plumes for the second and green or green/yellow plumes for the third (which is not exactly the same as the French line). For regiments in direct service to the Duchy of Warsaw, cord colour depended on whether that regiment had been issued French infantry dress or adopted uniforms based on national regulations. The plate I have is of the 9th Line, showing a fusilier and a grenadier officer, both with white cords (or silver for the officer) - which suggests that the grenadiers of the 9th, at least, had white cords to the bearskin (having adopted French-style dress). Unlike the Vistula Legion fusiler, that of the 9th has a white pompom rather than plume. Also in the plate is a voltigeur of the 2nd Line who has yellow shako-cords, just to muddy the water further. I have seen also some near-contemporary drawings of voltiguers with white cords (I think the basis of the Funcken uniform guides of the 1970s).
If you painted your grenadiers with a red plume and white cords, you certainly wouldn't be wrong. I suspect you would be equally 'not wrong' if you painted the cords red. Duchy of Warsaw foot artillery certainly had red-corded shakoes.