What are the possibilities of a dwarf planet orbiting opposite Earth's orbit?

I am drafting a short story about a small object orbiting the Sun at the precise 'speed' as Earth's opposite the Sun, undiscovered by us due to its position. The problems I see with this include the following.

1. We would have detected it by now, because of the gravity that impacts other objects in the solar system. How small would it have to be (just an order of magnitude is fine) to not be detectable?
2. Would our spacecraft beyond Earth have 'accidentally' snapped a picture of it?
3. Must it reside at Lagrange Point 3 ($L_3$) in order to have our orbital period?

Side note: I was re-phrasing the question to go from "small planet" to "dwarf planet" and accidentally deleted the wrong word. Before posting I looked up to see I had written, "What are the possibilities of a dwarf orbiting opposite Earth's orbit?" and truly laughed out loud. That would have gotten some weird looks haha.

posted over 9 years ago

Mikey‭

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