Is the original Sword / Shield save compatible with the new DLC included cartridge?

[stringDTD]

Assuming the original Sword / Shield has the DLC enabled, am I able to get the save from the original cartridge (using a save editor) and use it with the new cartridge included version of Sword / Shield?

[sceptios]

Saves for Switch games are not stored on the cartridge. They are stored in the Switch's internal memory.

To answer the other part of your question, a SwSh game+dlc save will work with any combination of physical and digital game+dlc.