Could there be a silver badge for 2000 rep over the cap (earned but not received due to the daily cap)? My thinking is that extra activity, even after the cap is reached, is to be encouraged. I don't think this could be awarded multiple times, but I don't know.
My thinking is that extra activity, even after the cap is reached, is to be encouraged.
One of the purposes of the reputation cap is to get people to take a break from the site so they don't burn out; I don't think a badge for exceeding it makes sense. If they really wanted to encourage extra activity, they'd just get rid of the rep-cap, so I think your assumption is flawed.
There are badges for hitting it, but once you're 1 point over, there's no incentive to keep going (well, other than accepted answers and bounties). This badge would create an incentive to keep going indefinitely.
Additionally, you are encouraging "luck" which seems odd - one major reason people will hit a large amount over the cap is cause their post hit reddit or slashdot.
I don't see any reason to add badges for the reputation users would have gained if there were not a reputation cap.
Consider also that users can gain more than 200 points in a single day because of accepted answers and (if I correctly recall) bounties. If a user's reputation was daily increased by 300 in X days, I think that would be equally a good result; the user would not get any badge, apart the already existing ones, even with your proposal.
If you get 200+ upvotes in a single day, you will already get at least half a dozen badges like good-question, great-answer, enlightened, and probably a few tag badges as well.
This makes another badge look more like a badge-badge. A medallion to people who already have a drawer full.