For AE at least, the first sentence sounds entirely idiomatic to me, for conversation or conversational writing.
You could also say '... any more than he already has'; in fact, I might prefer that.
The second sentence isn't exactly wrong, but it seems less natural to me, more like a non-native speaker trying to be overly clear.
BE, however, has some instances where they add 'do' where AE doesn't. I'm not sure I can describe the entire category off the top of my head, but it has something to do with another helping verb or modal verb. I imagine it sort of like this, but BE speakers might be able to shed more light.
Has he finished the job / gone shopping yet?
No, but he will. (AE)
No, but he will do. (BE)
Does X have any relation to Y?
I don't know; it could. (AE)
I don't know; it could do. (BE)
But again, I don't know if that would even apply to your example.
PS: These questions would be perfectly legitimate in the Sprachlabor section, and might get a little more attention there. I wouldn't move this thread now, but a thought for next time. (-: