I watch the show "The Good Doctor". It was used in season 2 episode 4. In flashbacks, Shawn had with his mother/caregiver (couldn't quite figure it out), she was dying and he had to move. They used it three times in this episode. I know its a bird, but what is the idiom behind that?
Internet searches for "tough titmouse" produce very little of substance aside from discussions of episode 4 of season 2 of The Good Doctor, which is titled "Tough Titmouse" and initially aired on October 15, 2018. A recap of the episode on CelebDirtyLaundry.com indicates that the expression was used in a character's foster family, twice by the foster mother and once by the foster son:
The psychiatrist meets with Claire, Morgan and Dr. Lim, trying to figure out what is best for Kitty. Outside, Max apologizes to Shaun for hitting him and they talk about what do they think about when they are scared. Max says “teaser”. Shaun flashes back to his foster home, where his foster mom used the term “tough titmouse” and told him to get dressed and he’s going to go to school and face the horse’s ass!
Nicole learns her wounds will heal, but Alex tells her they know Max caused her injury. She insists Max has never hurt anyone, but when she sees the bruise on Shaun’s face, Alex says she needs to find a place for Max and there is no shame in finding help for her and Max. She fears Max will hate her. Shaun remembers the moment his foster mom tells him, he has to move out because she is very sick. Shaun tells her, “tough tit mouse!” Shaun is able to tell Nicole that he won’t hate her, he will just be scared. Melendez feels she will find the strength to keep going.
Nicole explains to Max what is going to happen, as Melendez and Alex find it difficult to watch. Shaun remembers moving out of his foster home, admitting he is scared to leave and she tells him, “tough titmouse!’ After he leaves, his foster mom tears up.
You can view the third instance noted above in a brief Facebook video excerpt, but be prepared for community-theater-level dramatic acting.
In each of these instances, "tough titmouse" doesn't allude to a proverb or other cultural back story involving a tufted (or any other species of) titmouse; it is just a quasi-euphemism for a cruder expression that is itself simply a dismissive expostulation. Presumably, the foster mother in The Good Doctor settled on "tough titmouse" as being more presentable than "tough titties," which, as Jason Bassford points out in a comment above, is (or was) a fairly common rude dismissive remark in parts of North America.
"Tough titmouse" is thus somewhat similar to the expression "tough snot"—which is a poorly disguised euphemism for the cruder but equally dismissive "tough shit." (In Texas in the 1960s, "tough snot" was fairly popular among English speakers in the 10-to-13-year-old cohort: it was rude enough to please a sixth-grader dismissing a peer's concerns or objections, without incurring the full wrath of any adult who might overhear.) The main difference is that "tough titmouse" is a bit cleverer because of its play on "tufted titmouse," a chickadee-like bird that lives in much the eastern half of the United States (and a bit of southeastern Canada).
Dianne Porter, a blogger at birdwatching.com posted an item called "Tough Titmouse" back in 2006, but her post seems to have nothing to do with the euphemism used in The Good Doctor. Instead, it focuses on the persistent aggressive behavior of a male tufted titmouse that lived near her house and spent months challenging mirror images of itself in her house's windows and her car's side mirrors and chrome trim. The wordplay here seems limited to tough and tufted.
A Google search turns up very few published instances of "tough titmouse," including just one in a book—in Kieran Crowley, Shoot: An F.X. Shepherd Novel (2016)—where it appears to rather ludicrous effect:
"That's off the record," I told Mel. "They don't want that out."
"Tough titmouse," Mel snarked. "If we got it, the Mail will get it soon. Frigate—we're going with the soaking story."
He was gone. I gave the bad news to Izzy, who relayed it to his boss, who went, well, ballistic.
I have no doubt that "tough titmouse" as a euphemism has some following in North America and may go back decades in some areas, but there is nothing more to it than the convenient similarity of the words to something naughtier—not unlike "particularly nasty weather" in the old joke.
a.) An insincere or sarcastic way of saying "that's too bad", or "deal with it"
b.) Short version of the phrase "'Tough titty', said the kitty, when the milk went dry."
e.g. Child: But Mom... I don't want to mow the yard! Mom: Well, tough titty. You're going to do it anyway.
c)small, active bird, expressing something small.