A biphonic language

I'll take a stab at the "evolve" part. Throat singing might be a possibility, so we could look at the environment surrounding that.

"The popularity of throat singing among Tuvans seems to have arisen as a result of geographic location and culture. The open landscape of Tuva allows for the sounds to carry a great distance. Ethnomusicologists studying throat singing in these areas mark khoomei as an integral part in the ancient pastoral animism still practiced today."

(https://en.wikipedia.org/wiki/Tuvan_throat_singing)

So in the context of conlanging, a landscape with sparse vegetation (or other... I'm not very familiar with geography) and perhaps combined with animist beliefs might explain the development of a polyphonic language, which would include not only specially spoken vowels, but also perhaps the use of false vocal folds and other throat singing techniques (https://en.wikipedia.org/wiki/Overtone_singing).

In addition, you mention the possibility of such a language being doubly tonal. In that regard, a humid area might be conducive to a tonal language, though the Tuvan language is considered a pitch accent language rather than tonal. (https://en.wikipedia.org/wiki/Tuvan_language). Alternatively, tone can arise from contact with other tonal languages.

In a way, vowels are already biphonic! Acoustically, vowels (and most sounds, actually) are simply combinations of formants: specific frequencies at which the vocal tract resonates. The differences between vowels are then caused by differences in the frequencies of the formants. This can be easily seen on a spectrogram, like this one from Wikipedia:

In this spectrogram, the vowel [i] has its first two formants at roughly 500 and 2500 Hz, while [u] has its first two formants at roughly 500 and 1000 Hz.

Now, it turns out that biphonic singing works on exactly the same principle! Biphonic singing works through shaping the mouth so that one of the formants becomes loud enough to be perceived as a separate note. Note that this uses exactly the same mechanism used in vocalising vowels; the only difference is that the shape of the mouth is changed slightly to emphasise the formants. To experience this yourself, you can try saying [u͡ʉ͡y͡ʉ͡u] very slowly; if you listen carefully to what you are saying, you should be able to hear a note growing higher and then lower in pitch. This note is one of the formants of the vowels (the first formant, I think); Biphonic singers simply emphasise this formant so it becomes loud enough to easily hear.

So, to summarise, languages already use vowels which could be described as ‘biphonic’ — the only reason we don’t hear it that way is because the formants aren’t as strong as they are in biphonic singing.

(For more information, you may find https://newt.phys.unsw.edu.au/jw/voice.html and https://newt.phys.unsw.edu.au/jw/xoomi.html interesting.)

You already hint a possible answer in the question: There is an art form named throat singing. A community where throat singing is practiced may carry over the some biphonic distinguishing features to their language. This may include borrowing a biphonic pattern for some formulaic expression from a throat song, or words with special poetic or religious meanings. Once established as a linguistic feature it may spread all over the language.

I think, linguistically this would be still described as a tone system, but with rather complex tones in it.

P.S. I once listened to a biphonic whistler who was able to whistle some classic music with two voices simultaneously. A flashing experience.