させていただけますか means してよろしいですか, and people use the latter more commonly. いただけますか is thought to be more polite, but it's also thought to be overly complicated.
させてよろしいですか is the common way to say something. It involves a third person. It's like what you would say if you wanted to ask, "Can I let my son play here?" -> 息子をここで遊ばせてもよろしいですか? It's kind of a way of saying させてもいいですか. In as much, you can also say: 息子をここで遊ばせても良いですか？ Probably most common of all is, させてもいいですか, which is used amongst friends, and it's also usually used with children. Sometimes people use it with foreigners, thinking よろしいですか is too much more difficult, as compared to いいですか.
させていただきましょうか is kind of like させていただきませんか, which means, "Shouldn't I check it," but it's not a good parallel.
You should think of it more like this, to avoid confusion:
見させていただきましょうか = Boss to employee: "I should check your work..."
みていただけませんか = Employee to boss: "Shouldn't I show you my work?"
ませんか is more polite, and you should use it when speaking to your boss. Bosses use ましょう to sound kind because they don't want their workers to feel like they are under the gun, when everyone should be happy in their workplace. It just makes things worse for everyone. Your boss would ask if he could check your work, but you wouldn't ask your boss if you should check his work, under ordinary circumstances. So, ましょう to ません isn't good for a transformation drill unless you also transform させる and する, respectively. Then, you could keep the same mental context while practicing.
