Sorry I don't know what balmex is - & would have thought that if the rash is specific to the genital area that bathing & then using a soothing cream daily would be the right thing to do. Perhaps change creams etc.
However, I would also think carefully & honestly regardless of the difficulty & emotional issues as to whether anyone could be interfering with your daughter. A rash specific to this area at that age & if she's kept bathed etc, has no allergies, but does have say daycare, preschool, babysitters or anyone who could touch her (that includes young children - who may be copying on her what is/was done to them!!). Believe me I do know how distressing this thought & reading this is, but she's your daughter & at 3 can not protect herself & may not have the knowledge or ability to tell you, so it is up to you.