Would "What can I bring to the dinner? " work as well.
I think it should also be accepted.
"What will I bring to the dinner?", is also an accurate translation of the sentence
More of a transliteration imo, that sentence doesn't strike me as very good English.
Isnt there particular equivalent for should would could in Portuguese?
"What should I bring for dinner" sounds more common for me.
The translation is "What should I bring to the dinner?"
Would this mean that the Portuguese sentence would be better written as "O que eu traga para o jantar?" where "traga" is the first person present in the subjunctive mood?
No, you can't use the subjunctive mood here. You could use it if you said something like "o que você quer que eu traga para o jantar?" (what do you want me to bring for dinner?).
What about *O que eu devo tragar para o jantar?"