You need to first consider the differences in laws between US and Japan when it comes to resumes.
In the US, it is against the law for potential employers to ask the applicant's race, color, country of origin, marital status, religion, age, etc. As the photos reveal the ethnicity, race, age, etc., it is discouraged (to the point of banned) to request a photo of the applicant, unless the job is for acting or modeling where the headshots are required.
In Japan, it is my understanding that in addition to the resume with photos, employers can limit the age and/or sex of the applicant for many positions. I don't think it is prohibited for potential employers to ask their applicants personal questions that are prohibited in the US.
That being said, and provided that Jersey Boy is a Japanese national, if Jersey Boy is seeking employment in Japan, it is expected that you submit a rirekisho no matter what. You may want to add an English resume with your rirekisho if it is requested or if you think it's beneficial. Unless you secure a position while you are in the US (as an expat), you'll probably need to go through what everyone else does in Japan.
You may want to attend one of those job fairs held in the US for college grads of Japanese nationals. I see ads in Japanese community papers and Front Line from time to time. You'll probably get proper information and more insights appropriate for you in such forums. It may be beneficial for you to get in touch with recruiters in New York City that specialize in Japanese companies.
Even if you don't have any working experiences in Japan, if you are seeking employment in Japan, well, you know what they say, do in Rome as Romans do. Until the law is changed to prohibit including a photo in your resume, we just have to deal with it, like it or not.