Bizarrely, https://en.wiktionary.org/wiki/by%C4%87 disagrees with this blanket rule of "być requiring instrumental". To summarize, być can apparently take a nominative if the object of być is well-defined. That suggests that "Jestem mama" might be correct in some circumstances. Jellei?
he already wrote about it here:
PART TWO AND A HALF: IF Y IS A NAME OR/AND A SURNAME
So sometimes in circumstances described by Podavus
And how do I say: "I am mom". [A mom striving for her son to recognize her].
Jestem Mama could be ok. (also "Dad jokes")
So, may I explore a little further? Jellei writes "Being "a doctor" kind of defines you, so it goes into Instrumental.", which I get, having already encountered "Dzień dobry. Jestem lekarzem". How about being "THE SPECIFIC doctor" whom, for example, I just called in an emergency, and has just arrived. Would that be a legitimate use for "Jestem lekarz"? Do the alternatives between Nom. and Inst. go some way to covering the gap left by not having specific definite and indefinite articles in Polish?