Yes you could do both, as long as you're not having a shed load of Calories, and 1 Fish Oil pill and about 150 calories of Protein and some Carbs, won't be busting your body with major Fat storage
Low Fat would be better, because regular , is laden with Fats, and might just contribute to more body fat, if you're calories at night are too high, for the resting metabolic state to adequately deal with.
Low Fat contains roughly 1g Saturate, and no Unsaturate per approximately 100g. I would even reccomened low Fat Cheese instead, as it's more Protein rich, and only has 2g Saturate and 2g Unsaturate per approximately 200g, (3.5 OZ).
Best to take Micellar Casein (80% milk protein, 20% whey protein) before bedtime as its a slow releasing protein, taking about 7 hours to digest. You can make your own by mixing Skim milk with whey but you are taking in to many calories as the the micellar has most of the carbs removed.
Every night before bed, I take one of two options:
1 scoop casein protein mixed with water & 2 fish oil pills
or
4 egg white omelette with 1/4 C low fat cheese and 1/2 C cottage cheese
Usually it's the first though cause I work second shift and I hate to cook when I get home.
I also do consume a protein shake at work occasionally on off days or even train days when I don't have time for snack. Sometimes I don't have time to break out of work environment to eat chicken, salad an veggie, so it's better for me to have at least a shake than nothing at all.
Couldn't live with out shakes! I just can't stand eggs and don't get me started on cottage cheese! But thats great nutrition advice. Sometimes I wish diet was more varied (I'm not the best cook!)