I agree that Satan was the serpent. We have confirmation of that in Revelation:
(Rev 12:9 NKJV) So the great dragon was cast out, that serpent of old, called the Devil and Satan, who deceives the whole world; he was cast to the earth, and his angels were cast out with him.
But where in the Bible is there any indication that Satan was the tree of the knowledge knowing good and evil?
You say that God did not mention the word "fruit" in regard to the tree of knowledge of good and evil. He does talk of not eating of that tree:
(Gen 2:16-17 NKJV) And the LORD God commanded the man, saying, "Of every tree of the garden you may freely eat; but of the tree of the knowledge of good and evil you shall not eat, for in the day that you eat of it you shall surely die."
What part of "every tree of the garden" were they to eat, if not the fruit? What part of the tree of the knowledge of good and evil were they not to eat? The bark? The leaves? The trunk? The roots? Eve certainly understood God to have meant "fruit":
(Gen 3:1-3 NKJV) Now the serpent was more cunning than any beast of the field which the LORD God had made. And he said to the woman, "Has God indeed said, 'You shall not eat of every tree of the garden'?" And the woman said to the serpent, "We may eat the fruit of the trees of the garden; but of the fruit of the tree which is in the midst of the garden, God has said, 'You shall not eat it, nor shall you touch it, lest you die.' "