Cow's milk cheese IS allowed on the MD, it just must be raw, aged cheese made from raw milk. You can find it at better health food and grocery stores.
Ironically, the one time my son tried raw, aged cheese, he felt it made him sick. Since then he will not eat cheese of any kind, but more due to MAP concerns...
Now, for the record, this is just my opinion here...
...Your question, "do you think that you either need to pick one or the other?"....I would answer, it depends. If you are in remission, I don't see why not choose from both diets (as long as you don't experience any adverse reactions from whatever you are introducing, of course!). If I was not in remission, I would stick to one or the other, and personally that would be the SCD for me. Again, jmo.
My son combines aspects of both diets and thankfully he's had great success. (He eats sprouted spelt bread, but that is the ONLY grain he eats. He does not like sweet potatoes, so that's out of the question! He has had maple syrup on rare occasions, but what's he gonna put it on? His spelt bread? lol! He just uses honey most of the time!)
That said, I think it's possible that there are some people who achieve remission on the SCD and, as Elaine puts forth, have to fanatically adhere to it to stay there.
If CrazyHarry is out there, he might have yet another slant on this!
I hope that helps a bit.....