4 Phase 5 Pin high speed motor driving (turbomolecular pump driver)
Not the easiest project you could find.
From my point of viwew it should be as follows:


use an unipolar stepper motor driver with a chopper. Complementary motor phases can be grouped, so you need only one current sense resistor for each pair, total two current sense resistors.
The driver has to accept ABCD input sequence, not a step/dir interface
The hall sensors have to be translated with a truth table to ABCD sequence
The above combination shall give you the equivalent brush DC motor behaviour with current limiting: max. acceleration and spinning up until back EMF voltage does not reach the supply voltage, or until the load torque does not reach the available motor torque set by current limiting resistors.
To be able to change speed you should introduce logical AND gates in series with the ABCD inputs and with additional PWM signal.




This is almost what you need. You still need to parse the hall sensors signals to the sequence of IN1...IN4, for speed control you can use a PWM signal on Enalble pin.

EDIT;

I am not an expert in the dicrete circituit built driver. IMO also pulse density modulation is not what you need. If you use a MCU then, from your 1st schematics add 2 current sense resistors grouped on Q1/Q2 and Q3/4 then you measure the current in both complentary phases. While Q1 conducts, the Q2 doesn't, same for Q3 and Q4. Implement two current PI controllers (for each phase pair) with PWM output. 

The hall sensors should provide the correct switching angle. Let's say combinations are as follow:


00 / 0 degrees
01 / 90 deg
11 / 180 deg
10 / 270 deg


You need to switch on phases in this pattern with respect to the rotor's flux:


0 deg / the resulting stator flux vector shall be 135 deg (phase 180 + phase 90)
90 deg / 225 deg (180 + 270)
180 deg / 315 deg (270 + 0) 
270 deg / 45 deg (0 + 90)
0 deg / repeat








simulate this circuit &ndash; Schematic created using CircuitLab