Welcome to Mi Yodeya Ryan! Thank you for sharing your insightful question - The Sefer Yetzira (2:3) is one of the earliest sources which discusses the connection between each group of letters which are interchangeable and explains:
עשרים ושתים אותיות יסוד חקוקות בקול חצובות ברוח קבועות בפה בחמשה
מקומות אחה"ע בומ"ף גיכ"ק דטלנ"ת זסשר"ץ [קשורות בלשון כשהלבת בגחלת
אחה"ע משמשת בסוף הלשון ובבית הבליעה בומ"ף בין השפתים ובראש הלשון גיכ"ק
על שלישיתה של לשון נכרתת דטלנ"ת בראש הלשון משמשת עם הקול וסצר"ש בין
השינים ולשון שכובה ושטותה] - Twenty-two letters are formed by the
voice, impressed on the air, and audibly uttered in five situations:
in the throat as guttural sounds; in the palate as palatals; by the
tongue as linguals; through the teeth as dentals; and by the lips as
labial sounds.
Although the letters in תְּכֵלֶת and שחֵלֶת certainly sound similar, when connecting words it's important to note which letters are interchangeable. In this case, although ת and ש are not technically interchangeable in Hebrew, Rav Hirsch would occasionally use those two letters interchangeably when connecting them to Aramaic terms in the Gemara where ת and ש are interrelated. Perhaps the more significant reason why it may be difficult to connect these two words is the difference between כ and ח. Though they do sound somewhat similar, the Sefer Yetzira groups them differently. Consequently, I don't recall Rav Hirsch connecting those two letters too much (if at all).