I think there is some truth to the idea that the formula is different, and the baby might have a hard time adjusting from one to another. I know that whenever I go to America, I have a hard time adjusting to the food, so I imagine formula could be different. My older brother said it took him about a year to completely adjust to American food after living years in Europe, too.
In your case, it's difficult. I think your opinion doesn't count much because you are a man. I'm sure most Japanese men wouldn't even have an opinion one way or the other on the matter.
Where is your baby going to live long term? If you plan on living with your baby long term in Japan, it might be best just to use Japanese formula as much as possible while in America, if only to keep the peace with the in-laws. However, if you plan on living with your baby in America for the long term, use the American formula by all means! It's too much hassle and too expensive to bother with getting Japanese formula over the long haul.
Just my opinion. Remember, my ex-mother-in-law hated my guts. (But she was a dragon lady anyway....)