In order to get control of both the rate of pressure change, and the absolute pressure, I would use a cascaded PID approach. You have an "inner" PID where the setpoint is the desired rate of pressure change, and the process variable is the measured rate of pressure change (the difference in pressure from one iteration to the next). The output from that PID is the voltage to send to the pump.
Feeding that PID, you have a second PID block, where the process variable is the current pressure, and the setpoint is the desired pressure. The output is the rate of change, which feeds the inner PID. You limit the range of that output to the maximum desired rate of pressure change. As the pressure approaches the setpoint, the rate of changes decreases and approaches 0.
Of course implementing this all at once is complicated, especially if you don't have much experience with PID. Start with the simplest case - a single PID loop with the desired pressure as the setpoint, the measured pressure as the process variable, and the pump voltage as the output. If that works, add in a setpoint ramp and see if it works well enough to get you the rate of pressure change you want - if so, that's simpler than cascaded PID.
Do you have the PID toolkit? If so, setting up a simple PID loop is easy. It sounds like you already have the other key parts - you know how to write a voltage to the pump, and to read the pressure. Hooking up the inputs and outputs to the PID block is simple. You'll then need to tune the PID, and there are many explanations of how to do this both elsewhere on this forum and on the internet. I can't recommend a particular tuning method without seeing how your system responds, but most likely an open-loop method such as Cohen-Coon will be appropriate here.
