I myself don't think that Bella is strong (quite the opposite) and found her trip to the ballet studio to be based more on stupidity and giving in to her emotions than strength.
I think Bella being gullible is pretty much canon at this point--Edward even comments on it in Eclipse, though his explanation is that Bella herself is such a bad liar that she can't tell when anyone else is lying. However, the ballet studio is something I give her absolute absolution for. Yes, what she did was dumb, emotional, and poorly thought out, but in my opinion most people in a situation in which a loved one is in immediate danger do not think rationally. This is hardly exclusive to Bella, and while obviously it would have been easier to just tell Jasper and Alice, Bella truly believes that James has her mom. And she also believes that her mom's only chance is for her to trade her own life for her mother's. Bella even realizes, in the book, that her mom might die anyway, but believes that if she does nothing her mom will definitely die, so rationalizes that she has to take the chance. And she knows, for a fact, that if she tell Alice and Jasper that chance will be lost to her, because they will not let her go. Everyone who is not Bella is, quite frankly, willing to sacrifice Bella's mom's life to save Bella's, and she knows that, so I don't think Bella is wrong in thinking she is on her own side in that situation.
It's not all that dissimilar to kidnapping situations where people are told that if they tell the police, their loved one will be killed. Sure, the smart thing would be to tell the police anyway--but taking that chance on a loved one's life isn't that simple for most people. And in this case, James will know for certain if it is Bella who is coming and will know if she is alone or not, so she thinks her options are limited.
That being said, throughout the series Bella tries to sacrifice herself for others and often winds up making life more difficult for the people she is trying to help. She does need to get over whatever complex it is that drives her to think that her pain and suffering is a commodity to be apportioned for the well-being of others.