There is some gravitational anomaly in our solar system. There was a paper from earlier this year suggesting that it might be caused by a small black hole, although I guess there is no reason it couldn't also be caused by a wormhole.
Edit: Here is the paper https://arxiv.org/abs/2004.14192. I misremembered it as a gravitational anomaly, but that is not completely inaccurate (it concerns the clustering of Kuiper belt orbits). The author (Ed Witten) proposes something similar to what you described for detecting the object.
Edit: Here is the paper https://arxiv.org/abs/2004.14192. I misremembered it as a gravitational anomaly, but that is not completely inaccurate (it concerns the clustering of Kuiper belt orbits). The author (Ed Witten) proposes something similar to what you described for detecting the object.