Sun's orbital elements
Strictly speaking the Sun's motion is going to be dominated by the gas giants and the planets and one asteroid is going to have negligible effect on it compared to those.

However if we assume there are no other items we can get an approximate idea of the effect.

You can use the similarity of the orbits.  The Sun ( in this simplified model ) follows a complimentary orbit around the center of mass of the two bodies.

The distances you have for the size of the obit scale like this :

$$r_1 = \frac \mu m_1 r$$

$$r_2 = \frac \mu m_2 r$$

where $\mu$ is the reduced mass :

$$\mu = \frac { m_1 m_2 } { m_1 + m_2 }$$

And as you can see $r = r_1 + r_2$ is the distance between the two bodies.

Now as you can see from those equations, the most massive body moves the least.

The mathematics comes from the way the standard two body problem is solved.

It might be instructive for you to compare the effect of your asteroid to that of Jupiter or Earth.