how to talk about genders with pre-teens

"She also asked why I think it is ok to be transgender and change your sex (with hormones and/or surgically), but it is not ok to identify as bigender."

The issue is conflating gender and sex. A person can't change their sex via hormones and surgery. It's not possible. They can modify their body to match more closely what they feel they should look like, ie they can change their gendered presentation, where gendered refers to stereotypes (e.g. long hair, smooth skin, breasts for a woman - a female with none of these things is still a woman).

Some people believe that they have a gender - the person who likened it to a soul has it spot on I reckon. That's fine.

It stops being fine when they impose that belief on other people who don't feel they have a gender, and don't believe that a man literally turns into a woman.

I don't believe that the communion wafer literally turns into the body of Christ but it doesn't affect my life in any way that other people do.

I don't believe that a male can become a female, but it does affect my life when they insist on access to single sex spaces based on their feeling of gender.

Bigender, trigender, pi-gender - all good. Dye your hair blue and wave a flag. Be happy, but don't compel me to share your belief system.

Probably won't make a blind bit of difference if your child is that way, then they're that way.

I'd go back to basics and look at exactly what the words mean. Sex and gender have different meanings.
Gender as a way to describe and define yourself has pro's and con's but not everyone has an innate sense of themselves in that way. Plus it's hard to seperate those ideas from sex stereotyping which is reductive and limiting.
Take it in small steps and encourage your DD to think around topics (not just this one) and not take everything at face value.

Bigender is someone who identifies with both a female and a male gender, I think?

That would be someone who embraces masculine and feminine aspects of their personality. This became mainstream by the 80s but now has become revolutionary.

People can believe they have a gender identity, although it seems to be meaningless when someone with a gender identity tries to explain what it is, but the problem comes when this is supposed to be like a soul that they are born with, supposedly having some kind of physical underpinning, such as being born with a pink brain.

I am dreading having these conversations with DC in future!

Could you draw parallels to religion?
E.g my DC knows that I am an atheist and when we talk about RE and different religions we talk about how some people believe in Allah, others in Jesus etc. Then I might say what I believe and ask what she believes etc.

Could you say that you support the right for people to have different beliefs and that no one should be oppressed for their beliefs. E.g that you are opposed to Islamophobia and supportive of muslims having the right to practice their religion free from persecution, but that you also don't believe in Allah, aren't a muslim and might not agree with the premises of the Quran. Equally you support the right of people to be transgender, you understand that some people have a gender identity (gendered soul/spirit) and this is an important part of their life and beliefs. You support their right to have those beliefs, to live their lives in peace and express who they are. But that you personally don't believe in gendered souls/spirits and believe more in science.

Obviously adapt according to your own religious beliefs!

I think perhaps it's useful to explain very clearly the difference between 'gender' and 'sex'?

'Gender' is really just social stereotypes applied to sex - by that token, it can indeed be limitless, there can be unlimited numbers of personalities. So long as this is clear, and it's not conflated with biological sex, there shouldn't be too many issues, I think. The problems arise when people try to blur the distinctions or mix up the two terms.

(Sex is very simple and straightforward - we are a dimorphic species, with two sexes).

I have no idea what 'bigender' is. This seems a somewhat arcane term for an 11 year old to be discussing?!