No, you need to change your output stage from a single push-pull amplifier into an H-Bridge, supplying one side of it with the output of the PID controller and the other side with the inverted PID output (which you can make by putting it through an inverting op-amp of gain -1, with it's non-inverting input biased at half your supply voltage, which Im going to assume is 12V/2 = 6V).
I see you've made the H-bridge already, but the signal for the 2nd side isn't coming from the right place.
So, with what I described above, when your PID output is 6V, both sides of the H-Bridge will be 6V, the voltage across the load will be 0V. If your PID output is 4V, one side will be 4V, the inverted one will be 12-4=8V, 4V-8V = -4V across the load. However, the H-Bridge would work better with PWM instead, which you can convert the analog voltage into by comparing it with a high frequency triangle wave https://www.ti.com/lit/ug/slau508/slau508.pdf?ts=1645171678473
Another solution, which is the simplest is to set the non-inverting input of the lower U5 opamp to half your supply voltage, then you loose half the range, but you can drive the load in both directions