I have been in this situation as the birth attendant believe it or not. I have only had 2 true shoulder dystocias since I started practice (I saw many more in training) and they have been with the same woman.
First birth was my 3rd on-my-own birth, so I will admit to probably needing more experience. The mom is a decent sized woman, with a generous seeming pelvis. She labored spontaneously, and pushed her baby out mostly side-lying, although did the dreaded semi sitting for the very end. She pushed the head out easily, but the shoulders wouldn't come. At that time, I was aware of only the more medical methods for managing shoulder dystocia. We did McRobert's maneuver, suprapubic pressure, then I used the corkscrew maneuver and finally freed the baby. Mom had a deep second degree laceration, baby was very slowly to start to breathe, blue and floppy, required bagging to start, and had a brachial plexus palsy (an injury to the nerves that supply the arm.) The palsy got completely better by 2 weeks of age. The baby was 9 lbs even, but had "linebacker" shoulders.
Three years later, I attended her second birth. After discussion of options, she preferred to try another vaginal birth. I felt more confident, having practiced the Gaskin maneuver, and having changed my practice to encourage moms to stay upright as much as possible. She again went into labor spontaneously, and pushed spontaneously. She mostly pushed in a true squat or on hand and knees, and stayed on her hands and knees for birth. I used every technique I've ever heard of to minimize sticky shoulders - hands off the head, no attempt to turn the head to line up the shoulders, no stopping to suction, encouraging mom to push as she felt the urge. The head was born very slowly, and then as the mouth emerged, it almost looked the head was sucked back in (called the turtle sign) I had not touched the baby at all to this point. Mom continued to push with all she had, but baby didn't budge. She was on her hands and knees already, and I tried to nudge the shoulders free, but couldn't. We had her move to upright kneeling (which is very like squatting) but baby still couldn't be freed. I finally managed to get a hand in and push the posterior shoulder free and the baby was born. His shoulder finally wiggled loose about 5 seconds before I gave up trying and tried to break his clavicle instead. It was over 5 minutes from delivery of head to shoulders. Baby was blue and floppy and required resuscitation again. He again had a brachial plexus palsy, but it again was better by 2 weeks of age. The mom had only a small second degree laceration from this birth. This baby was 9 lbs 8 oz, again with huge shoulders.
This mom and I have had a lot of discussion about what to do if she decides to have more children. We have agreed that unless we had good reason to suspect a smaller baby, she'd probably have an elective cesarean. We did have good outcomes both times in the end, but a minute or two more in each case and that may not have been the case. Babies do die occasionally from lack of oxygen due to shoulder dystocia. Watching her son be resuscitated last time was a significant source of stress and really ruined the lovely birthing environment we had going. Frankly, as her attendant, I don't want to be in that position again, watching a baby's heart rate go down and down and down and not being able to get him free.
And you cannot just wait until the head is out, see if the shoulders are stuck, and THEN do a cesarean. There is a technique called the Zhavenelli (I'm really not sure on spelling here) maneuver, in which the birth attendant attempts to flex the head and push the head back in for cesarean birth, but it is considered a last resort, and in many cases may not be possible. The head rotates and moves quite a bit coming through the pelvis and down, and it's really hard to repeat that in reverse. Even if the head could be pushed back into the abdomen, there is a high likelihood of significant damage the mom, and a high risk of lacerating the bladder or lower uterine segment trying to get the baby out abdominally. This procedure is considered last resort if the baby is going to die if you don't try something. I've never heard of it being performed successfully.
To the OP, it's hard to say what you should do. You'll have to follow your intuition, I think. I think in general, cesarean to prevent damage to the perineum doesn't have a lot of good research to support it. Cesarean to prevent shoulder dystocia doesn't either, though. I wish you the best in making a decision, and hope you'll find good support here no matter what you choose.