OK, a little progress here. I wrote my own UnrealScript wrapper for SteamUtils. I can get a text input window to appear when Steam Big Picture is active. I can only get Big Picture to work if I launch my game through the Steam client. I cannot find any way to launch the game with Big Picture activated, when launching from Visual Studio or a shortcut. Launching from the Steam client isn’t so bad, if I add my local development build exe as a non-Steam game, so I don’t have to upload every iteration through steamcmd just so I can test. But now it seems I can’t use my controller for some reason when launching through the Steam client as a non-Steam game. Ugh. There’s always something.
Now I need to figure out how to set up a callback to get the text when the text input window closes.
(Edit: Just to confirm there is a problem with the stock OnlineSubsystemSteamworks implementation, I tried using Steamworks.ShowKeyboardUI as shown in my first post, when launching from the Steam client. The keyboard overlay still does not appear even when Big Picture mode is active, when I launch from the Steam client.)