It does seem like some perfectionism and shame going on - it’s so hard when you’re in it but if you do change the name, it might be a quick comment for most people that you never even hear (mostly just “guess what?, Sophie is now Lucy” then more or less nothing) and maybe if you have a teasing family culture, maybe a couple of jokes. Some people will fumble it while getting used to it, but you can handle that, that’s normal.
If you keep Sophie, you will also be better at handling people making mistakes re her name - I don’t know how but fewer people say “Willow?” for Willa - probably because where she is most often now is repeated (school) but I also think I just don’t notice. She is her name now so I don’t and wouldn’t get thrown by a mishap in someone saying her name wrong. As you recover from pregnancy, the labour and first year postpartum and as she grows into her name, these things typically don’t feel as important. I know it feels different when you’re in it! Otherwise and anyway, would follow the above advice including by reminding yourself about why you liked Sophie in the first place, and really consider whether a name change would give you relief. if the pain of keeping it more than the discomfort of change, go for it! 