trans and do not even own a bra, let alone wear one; why? don't know; it just was never my "thing", never really thought about it so, don't have any reason or explanation as to why I don't wear one. I suppose that falls under the heading of: "If you've meet one trans person, you only know about one trans person."
Just like non-trans people, each of us is different and has our own unique quirks and habits and rituals. So, just because one trans person wears a bra to work but not at home, doesn't mean all trans people do this.
I think, the correct answer to your question, therefore becomes not to look at what trans people in general do, but instead to think about what your trans character specifically is most likely do do based of their personality.
So instead of asking:
I think sometimes people outside the lifestyle/community overthink who and what we are, and forget that we are just people. We feel love and sadness, joy, pain, anger, sorrow, etc, just like you. Some of us are shy and introverted, others are bold, outgoing and extroverted. Some keep to themselves and even their closest friends do not realize they are trans, while others introduce themselves to every stranger on the street as "I'm transgender!" Some are flamboyant. Some are drab. Some are steampunk. Some are goth. Some live in mansions. Some are homeless. Some have families who support them no matter what. Some are cast out and shunned by their parents and siblings. Because of the wide variety of backgrounds they come from, each trans person is very unique and very different and none of them fits neatly into a cookie cutter mold of:
"This person is trans, therefore they do ____"
I think one of my readers put it best, when they comments on my transgender main character, from one of my books, and they said to me: "I like how you treated him like a real person and didn't make the plot focus around his gender. I was able to read the story without constantly being reminded that he was transgender. I got to the end and it mentioned he was trans and I was like - Oh yeah! I forgot that."
I think that is something that makes a trans author stand out from a cis author, when writing trans characters. The cis author is pushing the trans-this and trans-that, and look at how different he is because he's trans, and I gotta point out how trans he is as often as possible, blah, blah, blah... because the cis-author is thinking:
"This character is trans, they are different, they are not my equal, I have to point that out as much as possible so my readers know I'm being diverse by including trans people in my books. I gotta point out how different they are from normal people by saying they are trans as often as possible and making sure I have them doing as many trans specific things as possible, so no one forgets they are trans."
Whereas the trans author, just writes the character, and the fact that the character is trans may, possibly, maybe, be mentioned once, maybe twice, at the most, if at all. The reason being because the trans author treats the trans character as just another character, just another person, and doesn't focus on how the character is different.
I think cis-authors, though well meaning, spend too much time focusing on how different a trans character is, when they should be focusing on how they are just people, just like everyone else and are NOT really that different from anyone else after all.
My suggestion is that rather then asking people what they do (because everyone is going to do something different) instead, ask your character what she would do.
What is her personality?
Is she comfortable with her natural boobage? The more comfortable she is with her natural shape and size, the less likely she'll feel the need to wear padding in private.
Is she self confident and feel the need to wear padding to feel good? She may have multiple bra sizes. Very large cup ones for out in public, clubbing, and dating, a less showy more medium size for work and class, and a smaller one for around the house?
Is she wearing those full chest, silicone fake boob style body vests? Those get uncomfortably hot and sweaty, and she's likely only wear then under bathing suits, or simply clubwear dresses.
Is she not a body conscious person, not worrying about physical appearances? Is she NOT trying to grow bigger boobs, because changing her body is NOT a part of her personality? Remember, not all trans people are trying to physically change their bodies. In fact, only a small minority of trans people do.
The media hypes up sex change operations, when the fact remains fewer than 3% of trans people undergo surgery; and the media hypes of hormone replacement medications as well when in fact fewer than 20% of the trans community every opt for hormones.
If your trans character is doing either, she is in the EXTREME MINORITY within the trans community.
Remember: transgender has been around since the days of Moses (Moses talks about it in the Bible). On the other hand, both sex change surgery and hormone replacements are NEW, still in experimental stages, still largely untested for long term results, people often die from them because they are so very untested for safety -due to them being so new, and probably the most significant question...
...if your character can afford to pay for either sex change surgery or hormone replacement therapy, what the heck is she doing in college? Someone who is enough of a millionair to be able to afford either procedure certainly has no need to go to college as any job she'll find from her college degree is gonna be chicken feed to the income she already has. Theory is not cheap, she's either a multi-millionaire or has taken out multiple mortgages against her home to afford it.
The reason only 20% of all transgender people ever undergo hormone replacement, is because only 20% of the trans community are mega millionaires wealthy enough to afford it.
This means, that your character already has some glaring inconsistencies if she's in college, yet can afford hormones.
This in turn opens up even more questions...
There exist, shady, back alley "scam doctors" clinics who offer hormones "on a discount". If she's in college, she's likely opting for some very questionable underground hormone "clinics"