I did in fact use 46 as my denominator in Part 1, and 47 in the other two. The reason I did that was because in Part 1, we don't just know our cards and the flop--we also know, because the K
is on board, that at least one card in Villain's hand is not a spade (making it slightly more likely that we hit one). So in that part only, I could reduce the denominator to 46, but once we expand Villain's range enough in the other two parts, we have to put the denominator back up to 47 since we no longer have even partial information about Villain's hand.
Remember when I said in the OP that there are some "very subtle oversimplifications"? This is basically what I was talking about. The real way to handle these range-adjusted probabilities I thought would be way too much work so I did it this way instead. I was wondering if anyone would ever catch it. Kudos.