Line 1a: 6...Nxd2 7.Bxd2 Bxd2+ 8.Qxd2 Qe7 9.Qf4 or 9.Qc3 with a slight-to-clear advantage.
Line 1b: 6...Nxd2 7.Nxd2 gives White a slight edge thanks to the bishop pair.
Line 2: 6...Bxd2+ 7.Nxd2 Bxd2+ 8.Bxd2 Nxe5 (8...Qh4 9.Bc3! Qxc4 (Not forced, but then what's the point of 8...Qh4?) 10.e3 Qe6 11.Qh5 with a clear advantage) 9.Bc3 with an edge. One possible continuation: 9...Qe7 10.Qd4 f6 11.e3 b6 12.Be2 Bb7 13.O-O followed by b4, with an edge.
I freely grant that the above isn't a refutation in the strong sense: the concluding positions in these lines are all playable for Black. But the real question is why Black would want to play such positions - there are basically no complications, and the positions are primarily technical. And if one can win such positions with Black against a peer or near-peer, one shouldn't play the Budapest, but technical lines instead!