You've already made decisions, but never the less, I'm going to stick my oar in. I think there is some (accidental!) verbal sleight-of-hand on this thread, and I also think the new applicant's trans identity is a red herring.
We talk about the right to female-only spaces a lot on this board. Those are places like hospital wards, changing areas, toilets, swimming sessions where the purpose is access to space and the equipment in it while males aren't in it. The personal beliefs of any other women present are irrelevant. Maybe the woman in the next cubicle believes the earth is flat. Maybe she's a trans activist who shouts at babies. Nancy "you're a sexual racist" Kelley of Stonewall UK could be in the next swimming lane and she'd have a right to be there! I think trans-identifying women have a right to use public provision for the use of female people, because they are female.
However, your group isn't a space; you're a private group of like-minded same-sex attracted women who meet up; the trans-identifying woman here isn't asking to access the table or that bit of the bar. She wants to be with your group, and you are people with freedom of association, not semi-sentient service automatons. You're not being funded by the taxpayer to provide support to any and all same-sex attracted females in your local area. For toilets, female-only means "all females welcome" because toilets are objects that don't care who sits on them. You are not inanimate objects and you meet up with same-sex-attracted women of like mind.
Returning to my mention of a red herring, this is where that herring is hiding. Nancy Kelley of "sexual racist" fame is absolutely female and absolutely a lesbian. But it would be perfectly reasonable for you to exclude Nancy Kelley from joining the group, because her values are so different from yours. How can you have a social evening with someone who freely calls you racists for your sexual orientation? That situation wouldn't be improved if Nancy came out as a trans man and started taking cross-sex hormones. It would mean she'd taken her belief in gender identity philosophy further, if anything.
So you're not excluding this applicant for being trans, as such. You just recognise, as with Nancy, that the philosophical differences between her outlook and your own are irreconcilable. Her trans identity is an expression of her beliefs, and those beliefs bar the way.
Some people would still call that discrimination, but they presumably also think it's religious discrimination that churches only employ Christians as vicars, and never employ practising Muslims. 
P.S. Cancel the cheque!