Would there be any difference in nuance if "jest" is left out?
No, not really.
Dziecko is nominative, so what case is ostatnie?
Also Nominative. "our last child" is one noun phrase and should be treated as such, if you changed the verb to let's say "szukać", which takes Genitive, then the whole noun phrase should take Genitive. "Szukamy naszego ostatniego dziecka".