The pass is actually 5719ft according to the sign at the top, and you do not start going up the grade until you are at the missile range HQ road on the west side of the Tularosa Basin, not at the 54/70 junction.
I drive 54-70 several times a year. There are no significant grades, but there are ups and downs around Corona, NM. Get your fuel in Santa Rosa and try to make it at least to Alamagordo before refueling.
I am not sure what mountains you are referring to on US 60, but I do not think you would have any trouble on US 60 either. Except for the decent into the Salt River Canyon it is fairly flat. Especially from Vaughn to Show Low where it starts down to the river.