Song Help Request Key of F riff for “…in a canoe” (Christmas Island)

[Oldscruggsfan]

Squarely in the category of “even a blind hog can sometimes find a hickory nut”, today I stumbled into what sounds like the correct 4-note riff for the “in a canoe” lyrics of “Christmas Island”. As with all tunes, I transposed to F in the hope of getting in the ballpark of my awful voice.
The strummed riff starts with the first fretted shape for Am7 (3542) then slides to G#m (same shape, one fret lower) and ends (final syllable of “canoe”) with Gm at the 1st fret.
Of course, what I’m describing is largely transitional. Yet, every Key of F chord progression I’ve found for the tune is written as going straight from G to C7 for the “in a canoe” lyric. That causes me to second-guess my aged and damaged hearing.
Does music theory / chromatic scale confirm the correctness of such a riff in Key of F?

 

[Renaissance-Man]

Hey,
Gm is a valid II chord in the key of F. And, it's common practice to walk a chord down 3 semitones for a transition. I don't know the song and I can't seem to find it by in a canoe. So, that's all I got.

 

[Wiggy]

... transposed to F in the hope of getting in the ballpark of my awful voice.
The strummed riff starts with the first fretted shape for Am7 (3542) then slides to G#m (same shape, one fret lower) and ends (final syllable of “canoe”) with Gm at the 1st fret.

Jumpin' Jims Happy Holidays booklet: Christmas Island in key F.

The song lyric "sail in with your presents in a canoe" begins and ends during a G7 (not a G). The G7 holds into a C7 to start the "If you ever spend..."

I'm not understanding where the Am7-G#m-Gm would fit in there.

 

[Oldscruggsfan]

I thank both of you for giving me two different perspectives, both of which were helpful.