There is nothing unreasonable about wanting to be out, and long-term closeted living can have very serious consequences, as you’ve already noticed: no social life, creeping anxiety/paranoia, a lack of external support, disaster in the case of a medical emergency… It’s no way to live, basically.

Try to reach out to other non-straight people, either in your area, or at least online. Try to learn from their perspective and experience on how to handle this. Spoilers though: they will tell you to come out, for goodness sake. Unless by “frowned upon where we live” you mean “we’d get arrested” or “we’d lose our jobs, ruin our chances at getting a mortgage and possibly lose the flat”, you don’t have a good enough reason not to. And so much to gain when you do.

As for how to convince her of this: it may be hard. Talk to her about your fading social life. You need friends, you can’t have only one person you can be yourself around, that’s a recipe for madness, not a healthy relationship. Get her to read all of “I’m from Driftwood,” or at least check out the It Gets Better project. Talk to her about maybe, therapy? Take her with you when you check out the local lesbian bookclub (yeah, not likely, but try to find something if you can).

Be compassionate, but be clear with her: you need to do this. You can come out slowly or you can come out all at once, but you can’t continue to live in the dark. It’s not a viable option for a healthy, fulfilling, beautiful life.