Sounds like you're hacking an existing footswitch-actuated peristaltic pump and want to turn it into a super automatic Pi pump with a footswitch override. An adventure. That's an expensive adventure given your pump, so let's be careful.
Just like the movies where they defuse bombs by finding the right wire (Tick tock. Tick tock) you'll need to find out what wires are connected to the footswitch. Look on the back of your pump at the green plug that goes to your footswitch. Pull out the green plug gently and get your multimeter.
A common solution for this situation is to have a normally open footswitch close when you push the pedal. We can test this hypothesis by using the multimeter to check resistance (ohm) of the footswitch. If we guessed right and the footswitch is normally open (NO), then your multimeter will show the resistance go to zero when you activate the footswitch. Even if you PARTIALLY push it, you should get to a point where the resistance goes suddenly to zero. If your multimeter shows different behavior, this is not the solution for you.
If we verify that the footswitch is a normally open, momentary on SPST switch, we can replace it with a RPi relay (e.g., Automation Hat). Remove the footswitch wires from the green plug and replace them with two wires that go to the RPi relay NO and COM screw terminal connections. Consult the relay vendor documentation for instructions on how actuate the relay and test your program. Test your program by making the relay click.
Now get out your multimeter again and test the green plug. It should behave exactly like the footswitch did when you run your program. If it did, go ahead and insert the green plug into the back of the pump. Run your program and the pump should turn just as it did with the footswitch.
At this point you can stop or go fancy and wire the footswitch in parallel with the relay to give you a manual footswitch override. This will give you Pi automation along with manual override.