... what might be the reasoning for naming the whisky "season(s)" and then "translating" it to the Kanji "time(s)" 時{とき}?
The word "time" can refer to many things -- the current moment, a span of time, a season, a time of life, several years. Just as in English, Japanese とき has various shades of meaning, one of which is 季節【きせつ】 ("season"). See the 国語大辞典【こくごだいじてん】 entry on Kotobank, specifically sense [三]③ (emphasis mine):
[三] 時間の流れの一部分、または一点をさしていう。
...
③ 時節。季節。時候。
More deeply, while the ultimate origins of the native-Japanese words とし ("year") and とき ("time") are uncertain, there is speculation that they might be related. If you can read Japanese, the 語源由来辞典【ごげんゆらいじてん】 lists some of these theories in their entries for とし and for とき. I don't think the compound derivation for とし makes much sense, and it is clear that both とし and とき stretch back into the Proto-Japonic stage of the language -- the Ryūkyūan branch has reflexes for both terms. Japanese とし appears in Ryūkyūan mostly as とぅし (pronounced something like //tuɕi//; this is a regular correspondence between Japanese //o// and Ryūkyūan //u//), as we can see in the various hits at JLect, while とき appears as とぅち (//tut͡ɕi//; same vowel correspondence, plus affrication of the //k// before //i//, similar to what happened when Latin ⟨ci⟩, //ki// became Italian ⟨ci⟩, //t͡ʃi//).
I have also looked into the 旧字体 list given by Earthliŋ in his answer to this question here: Variations in the "same" kanji, how do you know which one to use? But there is no correlation between those two kanji.
No, you are correct, there isn't any correlation: the kanji 季 and 時 are wholly unrelated glyphs (characters, symbols).
Is that [the use of the "TOKI" reading] because "KI" would be just a 1-syllable name and thus thought inappropriate as a product name?
I'm sure that's part of it -- ki is indistinct as a name, and could be any of very many things. The reading ki for 季 is also never used as a standalone noun, so Japanese speakers who hear ki used as an independent word would never think of 季.
The word とき is unambiguous, more distinct as a standalone noun, and is clearly tied to ideas of "time". Spelling it with the 季 kanji explicitly refers to the "season" sense, and adds a kind of poetical association to the name.
