I also do not wish to add legitimacy to them at all, as the way they put their style's name together is rather clunky Japanese, as was stated above. However, could they be using the "ka" reading of 家 for their style name, as it is pronounced "ka" in such words as 家族 (kazoku, which means family) and that kanji can carry the meaning of house? Just a thought, but still utterly ridiculous to think that they have legitimate lineage tied to Japan in any way.
Joshua Krill
Bujinkan Budo Taijutsu
"First footwork, then spine work, then handwork..."
- Shiraishi Isamu Sensei