I think it's more naturally to me say 'a bit'. I don't know why 'for' is before the sentence.
'A bit obvious, perhaps, but anyway it would get rid of her for a bit.'
There's a difference between a bit and for a bit:
Ex: I can speak with you, but only for a bit. <for a short time>
Ex: I only wanted a bit (of salad), but the server gave me an entire bowl. <a small quantity>
Does that help?