I haven't seen any research on this. I would have a lot of questions if they linked birth trauma to BPD or other mental illnesses. My son has some sensory processing issues (prevalent with people who are autistic and/or ADHD) that laypeople have tried to link to caesarean section. Except my son was not C section. He also has asthma, even though breastfeeding advocates say break milk eliminates allergies/asthma. Except my son was breast fed for a year.
Researchers would need to look at all forceps deliveries and follow them for years to see if any mental health issues arose. I'm willing to guess that there are a lot of forcep-delivered babies who do not have BPD or other mental health issues.
Going through the birth canal is a really intense thing! Even without forceps, a lot of babies born naturally have misshapen heads, and some of them have irregular shapes for up to a year, then end up fine