If you wanted them to be written as a character that lives normally like a cis character, then just write them as if you would a cis character. Normalize it, I guess? They wouldn't really have "wrong bathroom" moments if the society you write is not transphobic and your character lives life normally as their intended gender, right? They'd just go to their appropriate bathroom. You may not even have to mention bathroom visits at all. Are those scenes really necessary in the overall plot..?
It's like....
This is Joe. Joe is in the hospital today, because he's about to welcome a new life into his family. The family congratulates Joe as he holds his newborn in his arms in the hospital bed, after a somewhat short labour.
I don't have to write that he's trans or anything. Joe is Joe. Joe also just happens to have an uterus that he never removed.
I'm not a writer, but I do prefer that characters fit in so well that being trans doesn't need to be "explained" or explicitly said.
In regards to real-life experiences, I could say that, in an accepting society, it's just that any events are simply not questioned or not very "tense". For example, I had a friend who said they wanted to be referred to as they/them. Everyone went with it, no questions. A year or 2 later, they officially went by he/him pronouns. I was confused, but I just went with the new pronouns. Other people did the same. There were no arguments, no nosy business, nothing. Announcements were made, and the people just went along with it.
Similarly, another friend was born AFAB, but went by he/him pronouns when I met them. I just went with it, since all our existing friends did too. Years pass, and that friend switched to more neutral they/them pronouns. Same deal, I just call them by what they wanted. There's no extra effort on my part, really. Other people also did the same. We just... call them by whatever they wanted us to call them.
As for bathrooms, I identify as nonbinary/gender nonconforming/you get the deal. I don't really have bathroom problems. I either pick the universal (all gender) washroom, the single stall washroom, or the washroom that matches my genital needs. It may have just been a history unique to me, but my high school had some gender nonconforming students and often students see a "boy" go into the girl's washroom, or a "girl" go into the boys' washroom, but no one said anything about it. We were confused little high schoolers, but in the end we just went to do our business and didn't really care who was hanging around in front of the mirrors or stalls. I guess this would be your non-transphobic real life anecdote. I can't say that my entire city is non-transphobic, but we're fairly open (eg. having all gender washrooms available in most public sector buildings) and even if people are weirded out, they tend to stay silent or keep their thoughts to themselves.