I don't think people have been bitchy.
But I do think that some have lost sight of the fact that OP has just had a baby, and is therefore a bit vulnerable, and so some comments (such as those that imply that her child will be seen as 'an imposter', or that SIL will love her that bit less) are bound to be disproportionally upsetting to her.
Who wants to feel that they've made a terrible error of judgment in one of the very first, and most lasting, decisions of parenthood?
OP, to be honest I can understand why your SIL is upset, and I think that if you can bear to do it, you should really consider changing the first name. But if you're determined to keep the name as it is, I think that you should try to have a conversation with SIL in which you acknowledge that you 'get' why she's upset, tell her that you have always imagined having a daughter named Eve, and it was never your intention to upset her or to undermine the 'specialness' of your beloved niece's name. The worst thing you can do is approach her with the attitude that she's silly or unreasonable to be upset or that it doesn't matter at all. It clearly does matter to her, and you should be sensitive to that.
Congratulations on the birth of your daughter. I hope that the drama over the name doesn't take too much from what should be a very, very special time in your life.