Frequency domain design of a PI controller for a first order system with transport delay

Let's say I have following first order system with transport delay

$$G(s) = \frac{4563}{s + 64.77}\cdot e^{-0.000455\cdot s}$$

My goal is to design in the frequency domain a PI controller in the form

$$D(s) = K_p + \frac{K_i}{s} = K_p + \frac{K_p}{T_i\cdot s} = \frac{K_p\cdot T_i\cdot s + K_p}{T_i\cdot s} = \frac{K_p\cdot(T_i\cdot s + 1)}{T_i\cdot s} = \frac{K_i\cdot(s\cdot T_i + 1)}{s}$$

for that system so that I have \$PM\geq 70^{\circ}\$ and \$\omega_{BW}\geq 691\,rad\cdot s^{-1}\$. I have followed these steps in the design:

1. Bode plot of the\$G(s)\$

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2. Crossover frequency\$\omega_c\$

To be able to achieve \$PM = 70^{\circ}\$ I would need the magnitude intersects the \$0\,dB\$ at such a frequency (\$\omega_c\$) at which the phase equals \$-110^{\circ}\$

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Based on the Bode plot above I have \$\omega_c = 926\,rad\cdot s^{-1}\$.

3. Integration gain\$K_i\$

Based on the Bode plot above I need to set the \$K_i = 10^{\frac{-13.6}{20}} = 0.2042\$ so that the magnitude intersects \$0\,dB\$ at \$\omega_c\$.

4. Integration time constant\$T_i\$

Let's say the PI controller has following transfer function \$\frac{(s\cdot T_i + 1)}{s}\$ (I think it is justified because I have already found the \$K_i\$ value in the step 3.). The aforementioned transfer function reduces the phase at its breakpoint \$\omega = \frac{1}{T_i}\$ with value \$-45^{\circ}\$. Based on that fact I have decided to set \$\frac{1}{T_i}\$ a decade below the \$\omega_c\$ i.e. \$\frac{1}{T_i} = 0.1\cdot\omega_c = 0.1\cdot 926 \sim 90\,rad\cdot s^{-1}\$. Based on that \$T_i = \frac{1}{90} \sim 0.01\,s\$.

5. Design verification

I have created Bode plot of the \$D(s)\cdot G(s)\$

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and I have found that the crossover frequency has been substantially reduced in respect to the designed value \$\omega_c = 926\,rad\cdot s^{-1}\$. It means that I haven't fullfilled the requirement regarding the speed of response which is given by the \$\omega_{BW} = 691\,rad\cdot s^{-1}\$.

It seems to me that the problem is in the inappropriate selection of the breakpoint \$\omega = \frac{1}{T_i}\$ location. I think that I should take into account also the breakpoint of the integrator at \$\omega = 1\,rad\cdot s^{-1}\$.

Can anybody give me an advice how to proceed in the design of PI controller in my case? Thanks in advance for any suggestions.