What is the ruling on allowing my wife to be photographed without a headscarf, which is a condition for obtaining a travel visa to work in a European country?

It is not permissible to give the wife a picture in an immodest dress to obtain a work visa in disbelieving countries, because honor is not something that can be valued with money, and because residing in disbelieving countries may be unsafe for one's religion and morals. One should be content with what Allah has provided for them in their homeland.