High Resolution Gyro Question

yes, kind of. You need to be careful about when you memorise the previous values and handle CTS

may be something like this would be "cleaner":

#include <LiquidCrystal.h>
LiquidCrystal lcd(10, 9, 8, 7, 6, 5);       // put your pin numbers here

const byte lcdContrastPin = 11;   // LCD CONTRAST PIN
const byte CTSPin = 2;            // Clear To Send PIN
const byte RSPin = 3;             // Request Sample PIN

byte DMM_ARRAY[6];                // Where to store the Bytes read

bool Get_data() {
  bool success = (Serial.available() >= 6);
  if (success) Serial.readBytes(DMM_ARRAY, 6);
  return success;
}

void setup() {
  pinMode(CTSPin, OUTPUT);
  pinMode(RSPin, OUTPUT);

  pinMode(lcdContrastPin, OUTPUT);
  analogWrite(lcdContrastPin, 50);    //LCD CONTRAST AMOUNT

  Serial.begin(1000000);
  lcd.begin(16, 2);
  lcd.print(F("GG1320AN21 TEST"));
}

void loop() {
  static float previousValue = 0;
  unsigned long previousMicros = 0;
  float deltaTheta;
  unsigned long currentMicros = micros();    // you could do this with millis() too, depends on the precision you want

  if (currentMicros - previousMicros >= 1000000ul) { // 1s in µs
    unsigned long deltaT = currentMicros - previousMicros; // that will be give or take 1s
    digitalWrite(CTSPin, HIGH);     // CTS Open
    digitalWrite(RSPin, HIGH);      // RS
    delayMicroseconds(50);
    digitalWrite(RSPin, LOW);       //RS

    if (Get_data()) {               //Read the Serial bytes coming in
      float currentValue = DMM_ARRAY[2];
      delayMicroseconds( 200 );
      digitalWrite(CTSPin, LOW);  // CTS Close

      deltaTheta = ((previousValue - currentValue) * 1.113065); // count * 1.113065   ARC SEC / SEC

      lcd.clear();
      lcd.setCursor(0, 0);
      lcd.print(deltaTheta / (deltaT / 1000000.0)); //3 decimal place values

      lcd.setCursor(0, 1);
      lcd.print(DMM_ARRAY[2]);// Print btye 3 (LSB)
      previousValue = currentValue;
      previousMicros = currentMicros;
    } else digitalWrite(CTSPin, LOW);  // CTS Close
  }
}

(typed here from your code, fully untested)