I hold both Irish and American passports

Hi I was born in Dublin and also am a naturalised US citizen, so I possess both Irish passports. I am travelling now on my Irish passport but can I submit a visa application for China using my American passport which is in Dublin? Naturally I dont need a residency letter etc for Ireland and on my American passport it says I was born in Dublin. In summary, can I apply for an L visa with my American passport (either by a agency or a family member) while travelling on my Irish passport?