  We introduce a notion of residual diffeomorphism covariance in quantum
Kantowski-Sachs (KS), describing the interior of a Schwarzschild black hole. We
solve for the family of Hamiltonian constraint operators satisfying the
associated covariance condition, as well as parity invariance, preservation of
the Bohr Hilbert space of Loop Quantum KS and a correct (na\"ive) classical
limit. We further explore imposing minimality of the number of terms, and
compare the solution with other Hamiltonian constraints proposed for Loop
Quantum KS in the literature. In particular, we discuss a lapse recently
commonly chosen due to the resulting decoupling of evolution of the two degrees
of freedom and exact solubility of the model. We show that such a lapse choice
can indeed be quantized as an operator densely defined on the Bohr Hilbert
space, and that any such operator must include an infinite number of shift
operators.
