No combustion with fuel & water blow-by going on in the differential, so the differential oil temperature doesn't really matter. Different story inside an ICE.
The extra heat on the battery in the hot summer time without the cover doesn't do the battery any good. Heat is what reduces the life of a battery. The cover was put in the design by Ford as a thermal barrier more than a beauty trim piece. The cover stays on just fine without the fasteners...
The cover will stay on just fine without the 4 hold down fasteners - even at 100+ MPH, lol. I leave them out so I can just pull the battery cover off in 3 seconds if I need to get at the battery.
It helps a lot to keep the battery cool. Take a drive on a hot summer day and feel the battery without the cover. Do the same the next day with the cover on and feel the battery. Night and day difference in temperature.