I've been recently getting more into Warzone's singleplayer campaign, but I've found that I have to plan to play levels on one device at a time because singleplayer level progress doesn't transfer across devices for me (i.e moves I make on one level don't appear when I open that level on a different device.) Is there any way to transfer singleplayer level progress across devices?

Nope, the actual moves and orders from both you and the AI's in the game are handled locally (on the device you are playing from).

Which levels you won and skipped is stored on your account, so if you wanted to change to a different device, you should probably do so after finishing your current level.