Consider a Photon from Sun and travels with a velocity $c$. Now think we are that photon. For us, it looks like Sun is moving away from us with a velocity $c$. So, why don't we get attracted back towards Sun, because the mass of Sun would be infinite for us since it moves away from us with a velocity $c$.
You have completely mixed the modern and classical concepts of relativity. If you're talking about mass increment, you shouldn't calculate speed of Sun based on absolute time & space notion.
For you as a photon, space will be contracted to zero and time will be dilated to infinity. So, you can't calculate a speed (which is a time-like spacetime event) of Sun.
While its a nice satisfactory explanation, its not the real one.