Reading stuff online, I have largely confirmed that Ladder Only items do in fact drop in Single Player. However, I am unclear as to whether "Single Player" refers to the actual Single Player mode (playing alone) or whether it includes LAN play. Will the Ladder Only items continue to drop in LAN games (Other Multiplayer > TCP/IP)?
No, you'll need to use the Plugy add-on
Please note this add-on comes with a lot of quality of life features, so it'll look almost like a totally new game, but it's really the only way to play single player or LAN.
The features list is available on the site as well as the links for current/past diablo patches in case you need to go backwards.