Would the Pharisees go to Pilate and secure the tomb on the Sabbath?
Not likely. In John 18:28 the Jews would not even enter into the Judgement Hall unless they would be defiled. That was the day of the crucifixion. The Pharisees wanted to partake of the Passover clean so they refused to enter the Judgment Hall.
It seems unlikely they would defile themselves by going to Pilate on the weekly Sabbath. Also, the holy convocation (Nisan 15) that followed the day of the crucifixion was not a Sabbath so in this case the Pharisees would not be defiled. Matthew's chronology was different than the other three gospels and Jesus was not crucified the day before the weekly Sabbath.
The day of the crucifixion in Matthew was not the preparation of the weekly Sabbath like the other three gospels but the preparation of the first day of Unleavened, a holy convocation. On Nisan 14 the Jews prepared for the Passover by cleaning their homes of all leaven, crumbs included. It took some effort but probably didn't take more than two or three hours. It was still called the Preparation of the Passover.
The Day or Preparation of the Sabbath, however, was Friday.The word Preparation was translated a number of ways in those days and it meant Friday.
See the following where the capitalized words show where Preparation was found in the Greek.
The Didache 8:1 reads: “But as for your fasts, let them not be with the hypocrites, for they fast on the second and fifth days of the week, but do ye fast on the fourth and SIXTH days…” Kirsopp Lake’s translates the second and fifth days as Mondays and Thursdays and the fourth and sixth days as Wednesdays and Fridays.
Judith 8:6 reads: “and she fasted all the days of her widowhood, save the EVES OF THE SABBATHS, and the sabbaths, and the eves of the new moons, and the new moons, and the feasts and solemn days of the house of Israel.
Polycarp 7:1 reads: “So taking the lad with them, on the FRIDAY about the supper hour, the gendarmes and horsemen went forth with their accustomed arms, hastening as against a robber.”
II Maccabees 8:25-26 reads: “And they took their money that came to buy them, and pursued them far but lacking time they returned: For it was the DAY BEFORE THE SABBATH, and therefore they would no longer pursue them.
Antiquities of the Jews 16.6.2 reads: “and they be not obliged to go before any judge on the Sabbath day, nor on the day of the PREPARATION to it, after the ninth hour.”