"He has no option" or "He doesn't have an option" .... Could this work?
Well, for one Nie ma opcji or even better Nie ma takiej opcji! are quite well established phrases to refuse someone's proposal. Link to wikidictionary PL
He has no option - in theory it could be the shortening of On/ona/ono nie ma opcji. but I and probably the majority of native speakers wouldn't throw out the pronoun if we wanted to mean what you proposed, because then it confuses with the next meaning:
Są opcje - There are options, and negation of this is Nie ma opcji (opcyj in antic Polish) - there is no option.
3rd person pronouns are omitted less often than others, but it's still possible, especially that in real life language you would have some context. So yes, that could work and I added such options (he/she/it). Also "options", as the Genitive is the same in this particular example.
what about "There isn't any option." - says the same thing differently and works in English to convery the Polish meaning.
Well, it sounds like "Nie ma żadnej opcji", but I guess it's close enough, added.