Assuming your premises are correct...
1) The 10" Smallmouth is legal to keep from the Elkhorn
2) You may keep in a tank any legally caught gamefish in KY...

(I'm not certain if those are correct, but your premise also works in the context of living near Cumberland and keeping a 15" Smallmouth from some local creek, since Cumberland has an 18" limit)


Your Question:
How do I prove that I caught my 10" smallmouth in a place where it is legal?


The legal system doesn't ask you to prove you caught it legally. The legal system tells the authorities to prove you did not. So it is not on you to somehow prove you caught your fish from the Elkhorn, it is on the authorities to prove you did not.

CC