She's been FLARPING as the Boy-skylark class with Aradia, the local depressed goth tgirl with a big heart. Aradia says the class is particularly tough since only useful at high-levels, textually setting up the grindy, transformative element of it all.

Later on, Vriska (out, gatekeepy trans woman) might be dabbling in a little "cringe culture" by harming Tavros, specifically out of a desire to be "every Boy-skylark's worst nightmare". She might be roughly initiating her eggy sisters to try and push them out of the chaotic transitional st8 of figuring stuff out as fast as possible, so she doesn't get second-hand dysphoria and feel bad about her own passing.
Yes, Vriska 100% weaponizes her second-hand dysphoria on other trans women.

After Tavros gets paralyzed due to the major fall Vriska pushed her into, Karkat tells Tavros to "STOP PLAYING GAMES FOR GIRLS." which is an ironical joke about the fact that the games for girls (Tavros & Aradia were playing together) has turned into a multi-layered, dramatically interpersonal TO-THE-DEATH BATTLE-ROYALE FOR GIRLS.
The Gamer Girls have changed Gaming Pace, and from here Aradia wants to avenge Tavros's proportionally insane egg-crack Vriska forced upon her, causing some more tgirl-on-tgirl violence.

Another interesting detail is that the full BOY-SKYLARK song by rj lake directly narrates ACT 1. Refering to June by (dead)name, it follows up on all the themes of being "born strangely", being extremely clumsy & doing a mess everywhere (the goofy sylladex inventory shenanigans that very much mirror Tavros's clumsiness)as well as all of it not being too productive (Tavros's entire FLARP sessions and her love for useless side quests).
The song also hammers in on that oddly specific detail : the deadname "John" being somehow imposed at 13 years old, roughly around the start of puberty (if not a little later into it). This one detail has always been puzzling CIS READERS OF HOMESTUCK (demographic : small) but every normal person understands the implications of growing "too far" off the potential to choose your own name & gender presentation, especially around the PESTERSOME PROCESS OF PUBERTY. This is ALSO important for Tavros, as we will see later.

user Turnip (banging pfp my man) did well to bring up the parallels with June (and even Andrew Hussie herself, which I am the one to bring up here because I'm very silly). All three characters have links to the boy-skylark/man-skylark classes, both Tavros and June belonging to the breath aspect, they're all various flavors & temporalities of TRANSGENDER.

I think user Rei Mawura is also onto something, with the observation that Tavros obviously struggles with the space she takes up, which as OP pointed out is presented (comically so, but still) as a first physical disability she has to deal with. The little fairies having smaller horns is expressing her wish to get smaller, take much less space, which is then ironically defused by Vriska putting her in a wheelchair. That much is putting Tavros in a state where she has to accept her material body's limitations, and accept that she will be a BIG GIRL (for the horns mostly, but the wheelchair is also a heavy piece of equipment, especially if rockets are strapped to it. Not physically disabled myself, I am only assuming things here & this is not my best conclusion whatsoever.)

This entire thing about size might also be mirrored with June's old clothes she's wearing in Beyond Canon, too small for her grown-ass woman body, but she doesn't care much. Please don't let Vriska see this she will start on yet another major wave of tgirl-sanctioned transmisogyny.

Anyway good post OP