Should I omit "that" or not?

I think she has a better chance of being promoted than you!

"What do you mean? THAT I'm less smart than her?"

"What do you mean? I'm less smart than her?"

Should I omit "that" in the question above?

You can use that or omit it. Both are valid and there's no difference in meaning.

However, it really should be:

"What do you mean? [That] I'm less smart than she?"

But even that sounds a little too formal, so I would say:

"What do you mean? [That] I'm not as smart as she [is]?"