It should automatically save the position of the character in the save file so it's probably something to do with how you have setup the at beginning of scene actions.
Failing that, then sure you could use Lua script to store the coordinates inside of some vs values.
Stop character, before changing to menu scene. Then with Lua you could store the coordinates in some values like so...
Values["tom_x"].Int = Characters["tom"].Position.x
Values["tom_y"].Int = Characters["tom"].Position.y
In your at begin of scene action you can now check if character tom is on the scene & if they are then teleport them to the position in the values - I'm sure you can work it out from my code block above & then you can set their destination position.