Don:
According to Siegfried Breyer ("Battleships and Battle Cruisers 1905-1970") "the Only capital ships of World War I were fitted with anti-torpedo nets. They were removed after the Battle of Jutland and not fitted again." He states that the Konig recevied 10 hits during the battle and was repaired at the Kaiserliche Werft, Kiel, with the work being completed by July 21, 1916, so it would seem that the ship was not very seriously damaged during the battle (compared to the Seydlitz which barely made it back to port and was out of action until September, 1916).
In the general notes for the class, Breyer states that "All ships, with the exception of Kronprinz (Wilhelm) carried the thin pole mast until 1917... This was later replaced by a thicker one which carried in the top a larger control position with a RF (only Konig carried two control positions, one above the other)."
So I think that you are fairly safe in omitting the torpedo nets to depict the ship as she appeared from July 1916 until the pole mast was replaced in 1917.
I think that Breyer is as reliable a source as you could want.
Good luck with the build!
Mike