Perhaps you could clarify, do you think there is refraction when looking close to the horizon?
Of course. It's just entirely impossible for it to cause a magnification effect on the Sun, regardless of the Earth's shape.
I was working towards understanding another issue, not strictly related to the apparent size.
The refraction of sight lines close to the horizon can't occur on a flat earth. ( Excepting unusual ducting and temperature gradient effects )
The refraction of horizontal rays requires the curvature of the earth for the light path to go through air of different density.
The flat earth answer I imagine is going to be along the lines of the sun is not on the horizon, but somewhat elevated, so the sight line is not horizontal, and angled through layers of different density and therefore can be refracted downwards. Which would have the effect of making the sun appear higher in the sky, not lower. So on a flat earth it's the wrong way around.