Based on this model, the Sun's sibling most likely escaped and mixed with all the other stars in our region of the Milky Way galaxy, never to be seen again.
We would easily detect a star closer to the Sun than our known stellar neighbours. Any former sibling star further away from us than that would be affected by the gravity of other stars more than by the gravity of the Sun.
How far could the star have travelled in 200 million years? I find it hard to think it could be more than a dozen light years away. And the list of stars within, say, 20 light years, is finite.
I think you may be confusing the age of the Sun with the time it takes to complete one galactic orbit.
The sun is more than 4 billion years old while 200m years is a bit less than the period for one orbit of the Galaxy (225-250 my). The Sun has completed more than 20 orbits in its lifetime. Even a 0.1% difference in galactic orbital period between the sun and it's twin would put them 3,000 light-years apart by now.
Based on this model, the Sun's sibling most likely escaped and mixed with all the other stars in our region of the Milky Way galaxy, never to be seen again.