I'm of two minds about this.
I also have an unsupportive woman. As in the type who says, "Caitlyn's a freak. You're not, but..." (Jenner, in case you're wondering. Let's not debate on that, it's OT here.)
So... She doesn't want me to have breasts; but knew before she moved in that I was TG. That was a decade ago, and she's "so supportive" - I can finally sleep in a bra without her flipping out.
So, from my perspective, I see nothing wrong with, following a full disclosure and no marriage, doing things on the sly. That's in my case; she knew I had taken hormones, knew I wanted to be feminine, and knew I could be manipulated. Turnabout, etc.
But you indicate yours won't accept, either. And you indicate girlfriend, not wife. Not fiance, either?
No commitment yet.
And her family - I was born into a Roman Catholic family. I was caught a few times. You can guess where that went...
So - seems to me it's VERY risky.
OTOH, can you maybe make her more comfortable with your breasts? Make it foreplay? Maybe she'd come around that way...? But maybe not, and it could out you to the world. So what would the fallout there be? (I'm in a state where there'd be minimal professional fallout, and we have NO social connections here, so... That's not an issue for me. But other places, especially your home town, where you grew up and everyone knows you? Might be a reason to rethink.)
You also don't mention other things that might apply. E.G., you said you were flat-chested. Were you skinny? Do you have hips? Low body fat %? These things can make a difference, too - if you're fat like me, you'll look like a guy unless you really DO have hips - so public might not work. But maybe you could make it a game, and see where it goes?
Or, there are shows that could work out to your advantage, "It was what was on." E.G., "I am Jazz" (TLC) and "Becoming us" (ABC Family. Late transition, not as good as "I am Jazz," either. Choppy editing, objective seems to be to show the son is a douche, and the fathers are trying to meet all their obligations while being as femme as possible, without breaking anyone. Should be available on demand. Preview them first, just to see them and know what's coming - even if you have to pretend to be surprised...)
Jumping tracks, but - back to my growing up for a moment.
Even before being found out, I had felt out my parents. Didn't have shows, obviously - didn't need them. Gay jokes went over their head (E.G., one of the Pink Panther films, Peter Sellers, a "woman" comes out of the men's room, he makes the comment, "Getting so you can't tell the difference any more." I had to explain it, and the Drag show he was attending, to them... It was over their head. Despite their enjoyment of "Victor Victoria". And most gender-bending in the films, even if NOT played off as pure silliness, was either "perversion" or Perversion. We're talking about a mindset where "Ferris Beuller's Day off" was a dirty film because of the shower scene, where he's talking to the camera... And I was 10, 11, 12?
Point being, you can feel everyone out a bit, using Caitlyn or Jazz or someone in the news (and bring it up either for or against, or let it go and see where it leads.)
Anne Vitale has papers on sexism in MTFs, and also one on how Testosterone seems to increase the dysphoria. (Can't link, the company is being dicks again.)
And then, there are options you can do for "medication" - online pharmacies and Vitamin shoppe, etc. Get T-blockers, see how you feel. Use it as a diagnostic. It's the second prong of the fork - see how YOU change, and decide if SHE is worth it. Because you might find that the dysphoria can be managed without transition; or, maybe she'll get nosy, like the "new you," and WANT more of that person.
Lastly, I'm pessimistic, or jaded. ;-)
Prepare to lose her. It's part of a Samurai mindset: Live as if already dead, and you can fight without fear on the battlefield. You won't die there, you're already dead...
If you're afraid of losing her, you probably will... If you can keep resolute and less emotional in talking things over, you will be in a stronger position, and present things in a better way. And if she still wants to walk away, you couldn't stop that anyway. You keep control of yourself, and offer her the choices, once you determine what you can and will live with.
Lastly, if you have any health issues, do your homework, as you can easily find meds that will cause the desired side effects.... ;-) whether gyno or just low T. If they apply, you can talk to the doc about them, and you don't need an excuse - it's the meds.
Think it through and plan for where YOU want to be, and you might well get there. [Unless she's like my GF, in which case, run the F away. BEFORE it's too late. the Alpha woman quiz someone posted here? I took it pretending to be her, and as me, and we're both Alpha with some beta characteristics. I'm more Beta than she is, which means, more aware of and caring about others. YMMV.]