At the same time, when I say these words as a native American-English speaker, I cannot detect any difference whatsoever in their pronunciation. The "t" in the phoneme "nts" seems to drop out entirely -- but if so, why are these words not true rhymes?
As John Lawler says, in practice the two words sound the same.
Rhyme used to be judged entirely by ear, but online dictionaries use rules and analysis instead. So their definition of a perfect rhyme may differ from the average person's.
If you insert a t-sound into ounce, without slowing down, you end up with. . . ounce! (And it's perfectly pronountsed!) You can separate the last two letters of counts to make the t audible, but in normal speech we don't.
Poets and lyricists use rhymes like counts/ounce with a happy disregard for the velic flap. Wordsworth, Browning and Tennyson used near rhymes like love-move, suns-bronze-once and creature-nature, which no rhyming dictionary would allow.
Walker's Rhyming Dictionary of the English Language - "devised for the rhymer, not the phonetician" - is a 'backwards dictionary'. It badly needs updating and is tricky to master but it does at least leave it to the reader to decide which word suits his/her purpose.
Rhyme is based on phonemic form. "Counts" is phonemically /kawnts/, while "bounce" is /bawns/. They don't end the same way, so they don't rhyme.
The pronunciations of /ns/ and /nts/, as opposed to the phonemics, overlap, since the difference between the phonetic [ns] and [nts] is a rather delicate matter of timing the dropping of the velum to let air pass out the nose. The two words can end the same in pronunciation, but phonemes govern perception. The ends of the words still sound different.
They sound verrry similar - but, there is a subtle difference. (I'm talking about UK English. I'm from London.)
And subtle differences are important in poetry.
Bounce - ends with a softer 's' sound. Like the 's' in 'secular'.
Whereas, counts, having the 't' and 's' on the end, becomes that harder, more 'zzz' buzzy sound. Like the ts in 'tsar'.
I'd be a bit like 'er whar's that?' It would grate on my ear a bit.
I might, as a poet and songwriter, feel a bit short-changed, if presented with 'bounce' rhyming with 'counts' in a poem.
I'd feel more fulfilled if bounce rhymed with 'trounce', or 'pounce' or 'ounce'.
If 'counts' rhymed with 'amounts', 'founts' or 'mounts', I would also feel quite satisfied. But it depends...
Which word is best to use as a rhyme depends on the artistic work. You can get away with dodgy rhymes - if the words in the whole sentence and the words around it, hang together in a way that sounds pleasing to the ear. And/or is intetesting/makes sense.
Sometimes the use of a particular non,-fitting rhyme is humourous, or alludes to something else, mentioned earlier. Or has a double meaning. All of which can allow you to get away with it.
His countenance was just, but his hair was all mussed, as with a single bounced thrust, he undid his truss, 'for have you I must!: - the words burst out in a gust. .
And then, on all counts, in varied amounts, on dour countenances he bounced, the taxable amounts, as arguments were trounced, yielding... much poorer counts.
The 'counts' are noble Counts as well as 'sums of money' in my example.