Does the "prestige" of a foreign degree really matter in Japan?
Usually, nope. If you want to be a translator (changing the language of written materials) vs. being an interpreter (helping with spoken language), you might want to consider going through translation training. Does either school offer that?
Also keep in mind that despite knowing Japanese, you will be considered a foreigner and might just be offered little more than an eikaiwa-like position within the company.
Companies nowadays, especially the small to mid-sized ones, are acutely aware of the need to have their employees use English, but they don't have the finances to pay for such training, whether within the company or outside. Having a person like you on staff would be a benefit, but keep in mind that the main things most of these companies need English for are the following:
* reading/writing email and faxes
* doing teleconferences or the occasional videoconference
* interacting with clients whether they are sales people or the persons to whom you sell
Are you willing to be roped into that, or are you shooting for the mundane solitary and ill-paid position of translating contracts and marketing documents from one language to the next?
Got a specific field of business in mind? Might make a difference.
at the present time I'm not really thinking so much about a specialized job like "translator" or "lawyer" that some sort of specific education exists for. I'm thinking about what sort of degree would help me get my foot in the door at the average company.
Just having a degree in a specialized field and knowing lots of Japanese are not a complete set of marketable skills. They are good, yes, but fresh college grads here are hired based on the reputation of the uni where they graduated, and companies shuffle them around for 1-3 years until they get plenty of exposure to various departments, then zero in on what promotion track they will pursue. Foreigners, I don't know. But, if all you have is the degree, and no work experience, I wouldn't expect to be treated much better. In short, what do you think you can bring to the bargaining table? I think you might be valuing yourself too highly at this point.