I think the issue is that English rhyme requires a near-match of the last stressed vowel and of all following sounds until the end of the word. So you can sometimes just match the last syllable and get a rhyme, but not always, depending on where the stress falls.
For example, "example" rhymes with "sample" but not with "people" because the stress is on the second-to-last syllable.
But "redact" and "exact" rhyme, despite only agreeing in the last syllable, because that's the last syllable that's stressed.
"Takesies" and "backsies" are both stressed on their first syllables, so they would need to match from that vowel on in order to be perceived as a natural English rhyme, and they don't.
Are you intending to say that there is a transitive property of rhyming? And that if that if A does not rhyme with B then there may not exist a word C that rhymes with both A and B?
If so, your ideas are intriguing to me, and I wish to subscribe to your newsletter.