How do I (Save object position)?

How to store the position of various objects after the exchange of layout, and return to the current layout?

Set the object to global:Yes.

Be aware though, that if you swap between two or more other layouts, it will move those objects to other layers (within the base layout) occasionally.

You can use load and save actions

https://www.scirra.com/tutorials/526/how-to-make-savegames

It does not work well with overall placing: yes.

I would not have another option to solve my problem?

The objects are moved, hence when returning from a layout to another object should have the same position that was left.