If you were not dry mounting your print to the rag board, I would say no, the backing does not need to be archival.

Since you are permanently attaching your artwork to the rag board the rag board is now artwork and you do not want the artwork in contact with non archival materials.

Granted, it will take a while so it depends on how long you want it to last. Many of Ansel's prints are dry-mounted and I can assure you no gallery is going to put his print against regular foam board.

An option is to use 2-Ply rag board in between the mounted print and the foamcore to act as a buffered layer that can be changed out over time as it gets contaminated. Not only will it get contaminated from the foamcore but from the wall you hang it against and from pollutants in the air.