The only way to exchange the X and Y axis would be to physically do it at the controller. Why would you want to exchange them? I can not think of any reason why I would do that.
My cnc is setup like that. (spindle is on the Y… for few reasons)
Want to avoid switching stuff if i want to engrave after a cut. Keeping things (easy) aligned.
I understand at some point there is a good reason to switch. However !
You really need to switch the Y off the spindle on your CNC to get it back to the Cartesian standard, otherwise you will be having all these problems forever.
All new software such as LightBurn are built on Cartesian and make machine to machine control standard as well.
Personally I would recommend standardised systems.