Ideal Discrete Time Hilbert Transformer with constant group delay
There's not much need for complicated calculations. If you have the zero-delay frequency response $H(e^{j\omega})$ as defined in the first equation of your question, the frequency response with a linear-phase (constant delay) is given by

$$H_d(e^{j\omega})=H(e^{j\omega})e^{-j\omega\tau}=e^{-j\left(\frac{\pi}{2}\textrm{sgn}(\omega)+\omega\tau\right)},\qquad \omega\in(-\pi,\pi)\tag{1}$$

where $\tau$ is the group delay in samples.