Using preferred pronouns falsely represents, to other people, your willingness to capitulate to the broader program of trans-activism.
Functionally, it signals your assent to the overarching political coalition in which the norm of 'respect pronouns' is situated.
But there is at least one major criticism of that overarching political coalition. And I think no one should be expected to (falsely) signal allegiance in any way to the coalition until it actually takes that criticism on board. 'Allyship', remember, implies a two-way relationship. If you are expected to be an 'ally', then you should have the right to negotiate what you will get in return.
Here's the criticism:
Point 1: A 'self-ID' policy or law means one which collapses the distinction between people who are 'women' according to empirically falsifiable criteria (what you can prove or disprove through your five senses, or, say brain imaging) and people who are 'women' according to their own claim to being women (and nothing more.)
Point 2: Analogously, an 'inclusive' definition of 'disability' is one which collapses the distinction between people with, say, broken spines and with serious mental illnesses including personality disorders
Point 3: Same thing with self-ID for sexual orientation. Who is 'queer'? How can you disprove it? You can't, because it's defined in such a way that empirical evidence is not needed.
Point 4: The 'diversity, equity, and inclusion' cultural belief system awards informal social prestige and support to trans status, woman status, disabled status, and 'queer' status.
Point 5: People who stand to benefit from the above also stand to benefit from the formal targets and accommodations of DEI programs increasingly common in Western universities, corporations, and schools.
Therefore, conclusion: We should expect 'inclusive' redefinitions of male/female, disability, and sexuality to incentivise personality-disordered heterosexual males to call themselves 'disabled queer women'.
And this is exactly what we see:
Of course anyone who identifies as a 'trans lesbian' may actually be a sensible, decent, and deserving person (and I'm not just saying that; look up Miranda Yardley and Debbie Hayton.) Their situation also seems in and of itself quite likely drive anyone slightly crazy: to be romantically and sexually attracted to the opposite sex while also having a strong desire to 'become' the opposite sex.
But this doesn't change the fact that the activists are pressing for policies which create perverse incentives to solve problems that could be solved in other ways. (One of the reasons given for self-ID laws is 'we'll be outed'. This is disingenuous, because the fact that being trans is a protected characteristic in several Western countries is evidence enough of how little trans people have to fear in the West from being 'outed'.)