I experience the exact same problem when adding an HDMI monitor and like you mentioned on github, going to the background tab of mate-appearance-properties helps but is not convenient.
As a workaround I use a script to load a background made for the extra monitor space (3840x1080 instead of 1920x1080) and use the "spanned" picture option to fit it properly when using two monitors . As I assign it to a keyboard shortcut, this is not too much of a hassle.
If you want to try it you might want to delete/comment the lines related to custom layouts as they're broken in 19.10 as well as the Plank related lines if you don't use the Plank dock.
Should you want the same background on both monitors and since you use 1920x1080 and 2560x1440 monitors, I assume you should use a 2560x1440 background with the 'zoom' picture option instead of 'spanned'.
Hope this helps.