Because it means "I work using a doctor".

You only use instrumental in describing people if the verbs are być=be, stać się, zostać= become.

After most verbs instrumental case means with=using.
I'm writing with a pen=I'm writing using a pen= piszę długopisem. "

That's a calque from Russian and absolutely doesn't work in Polish. People that don't know Russian could probably not even understand what you meant.

wonderful news jako does not have declensions.

It means "as" in context like in this sentence translation of as into Polish in Pons dictionary

I cannot think of a situation where you would use Accusative after 'pracować'...

"pracować jako" is a fixed construction that means "to work as", and it is one of the rare cases when you use Nominative for something else than the subject of the sentence.

"Pracuję jako lekarka" works, but it is not masculine animate. It is feminine (and animate, but it doesn't matter for feminine). "lekarz"/"lekarka" is among those professions where a woman can use either a masculine or a feminine version of the profession's name.

Lekarz can be treated as masculine noun or general term for a (medical) doctor. Lekarka is only used to denote a women of this profession. In this case a women can say either that she is „lekarz” or „lekarka”, men can only go with „lekarz”.