Just a brief note: the lectionary in the Episcopalian Book of Common Prayer already "skips over" some bits of Scripture those of a more progressive bent would consider offensive - such as the part of Romans, chapter 1 that talks about homosexuality. I've not seen a BCP for eight years, but I specifically remember noting that omission.
Former Christian here, raised Catholic, spent 6 years as an episcopal... now I'm UU with pagan/earth based leanings. My husband is Epsicopal. Just wanted to give you some Episcopal info...
Yes, they are a more liberal Christian denomination... so they will tend to be more accepting of alternative lifestyles and beliefs. And they aren't fundamentalist in the sense that everyone MUST believe the same thing about a particular tenant of Christian theology... there is this idea of a continuum of beliefs. However, they are definitely Christian. And while they will not turn you away if you don't believe in God, the Eucharist ritual is very centered on Christian scripture and is participatory in nature... meaning there are creeds/prayers/responses that the congregation says together and that reflect Christian theology. This is why I don't attend church with my husband. It doesn't feel right to me to participate in prayers and recite creeds that I don't believe in and I feel that by not saying them with everyone else, I am separating myself from the communal nature of the service... that's my personal take on it... if you try an Episcopal church, you may have a different sort of spiritual experience.
Also, I want to address the the question about the Episcopal church getting to pick which stories they emphasize. When you're talking about individual parishes, this really isn't the case. Episcopal Churches individually are part of the larger American Episcopal church... similar to the way Catholic parishes are part of the larger worldwide RCC (although the Episcopal church in America is democratic in that representatives are elected to go to regional and national conferences where they vote on church policy). And the American Episcopal church is part of the larger worldwide Anglican Union... so anyway, part of this organization is that the liturgy (church worship) follows the same format in every Episcopal Parish. And the church follows a lectionary, which is a three year cycle of bible readings, usually one reading from old testament and one non-gospel new testament reading, and one gospel reading, and a psalm. These are prescribed readings that every church follows... so that it doesn't matter which Epsicopal church you go to or where in the country it is. On any given Sunday, all Episcopal churches are reading the same bible readings... it's not a pick and choose sort of thing. Although, the sermons you would hear on the readings have the chance of being pretty varied. As for my child, I am comfortable with him going to church with his dad (and not just because it is his dad's religion)... because on a whole, the Episcopal church is very open and focused on being loving toward all people. The particular parish that my husband attends is GLTB friendly... a large number of our friends there are lesbians and have been lovingly accepted into the community (even though I've never heard a sermon openly speak positively about homosexuality), and there is one lady there who is transgendered and everyone is very welcoming to her. So basically, what I'm saying is that Episcopals on a whole tend to be very warm, open, and loving people. But they are definitely Christian, and the church services and religious education are going to be Christian in nature.
Having experience in both the Episcopal church and the UU church, I'm really going to recommend the UU church for an atheist family. I think that you will have a higher likelihood of finding a place that allows room for you to raise your children atheist, but at the same time explore a variety of spiritual practices and ideas... and also there will probably be plenty of opportunity for charity and community involvement.