Quantization is an art, not a functor, as comments reminded you.
Here is a way to start thinking about your problem. You may arbitrarily choose the quantization $$ \hat H=c \hat p {1\over \hat x} \hat p, $$ which has your classical Hamiltonian as its classical limit, and is manifestly Hermitean, among 37897 similar hamiltonians with this property.
Your proposal is not Hermitean. (Your other way has sacrificed linearity…)
Working in the coordinate representation, equivalent to the momentum one by Fourier transformation (the representation has little to do with quantization!), you have $$ -c\hbar^2 \left({1\over x} \psi’\right )’=E\psi, $$ which you might solve$^\natural$, and Fourier transform, as you have learned in elementary QM.
$^\natural$ Try $$ x y''-y'+b x^2 y =0 $$ in Wolfram α.