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
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,
ません isn't good for a transformation drill unless you also transform
する, respectively. Then, you could keep the same mental context while practicing.