What justifies simplifying the integrands in this Ampere's law problem?
"Is the constant current statement correct?"

Well, the problem states, 


  and a constant 1A of current flowing through it


so, yes, it is correct.

Is the rotational symmetry the reason for removing the flux density term from the integrand?

Yes. If the flux density changed wrt the angle of integration, it would not be radially symmetric. The two are equivalent.