2011 Nissan Rogue AWD hybrid idea

My Rogue's rear differential recently crapped out. As I was replacing it, I started thinking, "What if I remove this drive shaft and replace the drive coupling with an electric motor?"
I found two RV batteries. They are heavy but 250 Ah might do the work of the AWD function at least. If I can make a relay be switched by my AWD control that normally engages the drive coupling from 0-6 mph, I would have just a little start assist and good low speed traction. I could even use two motors if I put one on the transfer case.
I am hoping for more efficiency but I am not expecting much. Please advise me on motors (the differential has a 3:1 ratio) and controllers.
If this is impractical I will probably upgrade my kids' go cart range.
Thanks