I think that any Double rifle that has an action big enough for a 500NE is too big for a .375FE. The balance of one or the other would have to suffer.

For an extreme example I have a friend, yes it it possible, who had a 600 Nitro. He had a set of 375FE barrels built for it. Because of the weight of all the extra steel needed for the width of the firing pins the 375 actually weighed more than the 600NE. I have to admit though that the 375 was a pussycat to shoot.