I may be commenting without a clue here, but I dual screen and my wallpaper engine is two separate wallpapers, one run on each. 

I haven't tried it, but there is no reason it wouldn't work if I swapped out the 2nd monitor for the smart TV via HDMI, as it is just acting as a display?

How are you trying to project to the TV?

I've always had similar problems with my TV that is about the size of a monitor, but it never displays any background I apply to it with WE. Not sure why but I guess TVs just don't work with it.

Technically, your tv will be seen as a new device. When you have a specific wallpaper on your pc, and change the display to another device, it sets to default wallpaper because you haven't specified "use this wallpaper for this screen".

Currently I run 2 monitors together that have the wallpaper duplicated, but when I disconnect them and use my third monitor (as a single), it shows a diferent wallpaper.

Hope I'm making sense here.

(This is all using wired connections to my monitors, so I don't know about a wireless screen)

Projecting to Smart TV