I've been trying to do something similar, but from a feminist angle. I've found that arguing on their terms in a way reinforces their ideology, when that is completely unnecessary. All you have to do is dismantle the ideology brick by brick, then there will be nothing left to address.

This is what I've got so far (sources not included):

Feminist Conspectus: Addressing the Trans Movement’s Ideology

The following is a succinct feminist perspective addressing the conflict between trans issues and women’s issues, by deconstructing the ideological narrative that has monopolised the discussion on this topic, as well as redressing the false representation of the feminist position.

The conspectus is divided into two sections. Section 1 Foundations of Gender targets gender specifically, its definitions, origin & more. Section 2 Assessing the Evidence tackles the claims made, taking the evidence in its entirety into account. The content of each is in the form of bullet points.

Section 1: Foundations of Gender

1a) Gender (synonym for sex): male/female

1b) Gender (construct): masculinity/femininity

2a) Gender identity: a social identity, not naturally occurring, but shaped by gendered upbringing in accordance with the norms associated with a child’s sex, not their preferred sex

2b) Cross-gender self-identification/felt-gender (almost exclusively incorrectly referred to as ‘gender identity’, & usually shortened to: ‘gender’): one of many, usually temporary phenomena collectively referred to as ‘childhood gender nonconformity’ that typically affects underage homosexuals.

• In the West this identity transition is considered proper medical treatment, indirectly pathologising homosexuality & making transgenderism into a method of conversion therapy

• Recently, the definition has stretched to include non-homosexual men with a paraphilia called autogynaephilia

3) Gender role (often shorted to ‘gender’): a traditionally masculine, or feminine social role, or performance

4) Woman (not a construct): an adult human female

5) Transgender men are adult human male transsexuals who present as feminine (names & pronouns, minimally) & most make their bodies appear female (e.g. tucking), but they share their sex, their gender identity & their privilege with men, which makes their preferred sex, felt gender & strong desire to appropriate women redundant:

a) Transgender men’s sex is unambiguous & binary, not on a spectrum, not assigned, not intersex, not female, but male. Therefore excluding them on these grounds does not indirectly exclude any women

b) Transgender men’s felt-gender is undetectable, indefinable & inconsequential:

• David Reimer was socialised a boy for the first 2 years of his life, then sexual abuse took its place under the guise of “re-socialisation”. So the major case purported to favour felt-gender theory, did the opposite. Conversely, the vast majority of infant males assigned female at birth & socialised accordingly did not desist

• Transsexuals prove that gender identity > felt-gender, otherwise they wouldn’t have to train themselves not to walk, talk, behave etc. in a gender-congruent way, nor would they have to put any effort in to emulating the gendered behaviours of their preferred sex

• Transsexuals’ gendered behavioural trends are consistent with their gender identity & inconsistent with their felt gender e.g. positions of power, media representation, crime rates, HIV distribution are male-dominated, while attempted suicide is female-dominated, regardless of trans or not

Section 2: Assessing the Evidence

• Proof trans people exist ≠ proof that "trans-womxyn are women"

• There is no academic source for the activist slogan: "trans-womxyn are women", not in the icd-11, not in queer theory, nowhere.

• By “brainsex”-logic gay men are women

• Trans-rights need to be tied to trans-status, not self-ID, otherwise we are throwing 2/3 of trans people (the female ones) under the bus (as well as 50% of the population)

• Sex-concordant pronouns are benign, accurate & descriptive, not "violence"

• Calling feminists "cis" & "TERFs" is violence by the trans movement’s own standards on self-identification

• ‘Male-exclusionary’ exists, ‘trans-exclusionary’ doesn’t

• Transgendered males benefit from male privilege (as with homophobia, transphobia does not offset male privilege), the ones who pass benefit from "cis privilege", meaning they are never more oppressed than women. Women in their natural, unfeminised state are “GNC” too.

• The male sex is the social majority, the female sex is the social minority, therefore “cis-privilege” vs. “transmisogyny” is a sexist reversal of the power dynamic

• The trans movement is just as sexist & androcentric as any other the microcism, male transexuals outnumber female transexuals in high positions & media representation

• The trans movement engages in parasitic activism, hijacking gay rights, BLM, feminism, the progressive movement etc. to further a biology-abolitionist cause

• There is more evidence of radical self-ID dogma harming women than there is of even the most radical of feminist sentiments harming transgender people

• Inequality between the sexes needs to be addressed independently by feminists without the trans movement vilifying & censoring them

• Women's sports, shelters, prisons, changing rooms, medicine & sexual orientations are not ideology-affirmation playgrounds for the self-ID cult. Two separate unisex spaces makes one redundant

• Being treated as a “woman” socially, among friends, family, neighbours, even shopkeepers is one thing, but being treated in law as indistinguishable from the female sex is sexism via erasure.

• Distinguishing between the sexes is justified, distinguishing between transgender-males who identify as women & transgender-males who identify as men is not, because differences in ideological beliefs are not material differences.

• Some trans people say “trans-womxyn are women”, others say trans-womxyn are men – neither view is "transphobic"

• "Gender dysphoria" in 2022 will be re-named, re-classified & de-pathologised. There is no mental disorder inherent to gender variance & therefore no symptoms, treatments, no clinics are required

• It’s unethical to transition kids when the majority desist before adulthood, without intervention

• The fact that the use of puberty blockers yields a 98% insistence rate is evidence of conversion therapy

• Homosexuality (exclusive same sex attraction) is overrepresented in the desisted, detransitoned & trans populations, therefore many trans people are just confused, or self-hating, gay people

• Transtrenders are teen girls who appropriate trans identities to fuel their teen sub-cultures

• Autogynaephiles appropriate trans identity to fuel their paraphilia (“transbians” only exist in the West)

• Transgender-males embracing feminine tendencies is authenticity, but emulating women, adopting the labels: ‘woman’ & ‘female’ even, is sexism via appropriation

• Gender abolition is feminism, men appropriating women is sexism via erasure

• Raising issues with the trans movement is not "transphobic"