I think you have something there. You could design a complex, but at least metastable orbit for an asteroid sized object that, in each period, would fly by both Earth and, say, Jupiter. Because it is metastable, only very small course corrections would be necessary to keep it going, and it could be arranged such that at every pass Earth gets pushed out just a little bit, and Jupiter pulled in. With the right sized asteroid, it seems feasible that this process could yield the desired results after billions of years.