Let's take my theory one step further. Tanomo is told that these skills appropriately match chi/qi and Tanomo writes the katakana version of aiki because it came from a Chinese source.
Takeda then is taught by Tanomo and writes aiki in katakana just like his teacher did. Why? As Takeda was illiterate but highly intelligent and had excellent memory, then Takeda would have been able to remember how Tanomo wrote it and rewrite it the same way.
Tanomo knew how to read and write using Chinese characters. He had no personal need to write in katakana, except perhaps as furigana (smaller kana alongside Chinese characters in a text) to indicate the mixed reading of the compound
(see Josh Lerner's post 36 at this link):
If we start from the documented premise that the reading is not a straight Chinese reading, but a strange mixed reading, the argument for the katakana being a sign that Tanomo was taught by a Chinese teacher is, well, strangely mixed.
Maybe he did, maybe he didn't, but I would have to suggest that this isn't evidence that is going to settle the question, or even point dimly toward a settlement.