Switch Control Between Two Different Techniques
the PID is looking at the RPMs and giving an output but that output is
  being ignored because another operation has servo control.


PID is not something, that happens somewhere in the background without you having any control. On the Arduino it is done in software, so you can totally control it. Why should you run a PID calculation, if you don't use the value? The solution is to simply only do the appropriate calculation depending on the RPM at any given time. No need to do 2 calculations at the same time. I will form that into a code with explanations:

You have 2 ways of calculating the needed servo values:


Your "getting close" method
The PID method


To switch between those methods you need to make a decision based on the RPM. If the RPM is outside of your interval, the first method is used. If it is inside, the second will be used. Decision/conditionals are often wrote with an if else statement. Look at this (not tested) code snippet:

int target_rpm = 700;
int half_interval = 20;
int current_servo_pos = 0;

void loop(){
    current_rpm = analogRead(AMPS);

    if(current_rpm &gt; target_rpm - half_interval &amp;&amp; current_rpm &lt; target_rpm + half_interval){
        // Do the PID calculation here and save the result in current_servo_pos
    } else {
        // Do the "getting close" calculation here and save the result in current_servo_pos
    }

    myservo.write(current_servo_pos);
}


I didn't go into detail about the actual calculation, since that is not important here. You may use any calculation, that you like there. For PID you can find libraries, that do the calculation for you, for example here at github.

The code will simply decide based on the RPM, what calculation should be used. If the RPM goes out of the interval for some reason (maybe is was disturbed and lost some speed), it will automatically use the "getting close" method until it is again inside of the interval.

Note: The numbers, that I used here, are arbitrary. You should change them to what ever you need.