I will take a stab at converting my comment into an answer, since there are no others. I do not think there is a single word combining both concepts, but there are words for each concept independently.
You clearly already know the first, for words which imitate a sound the word describes- onomatopoeia. From M-W:
: the naming of a thing or action by a vocal imitation of the sound associated with it (such as buzz, hiss)
The other part, where a syllable or partial syllable is repeated within a word, is called reduplication. Again M-W:
1: an act or instance of doubling or reiterating
2a : an often grammatically functional repetition of a radical element or a part of it occurring usually at the beginning of a word and often accompanied by change of the radical vowel
b(1) : a word or form produced by reduplication
(2) : the repeated element in such a word or form
From this blog I get a list of several example of reduplication, including flim-flam, riff-raff, and bye-bye.
Within this same list, I find a few more that might fit both criteria of onomatopoeia and reduplication: knock-knock, clip-clop (of a horse's hooves), tom-tom, and a few others possibly on the fringe, such as choo-choo (usually a name for a train, not the sound of a train, but originating from that sound). In other words, yes, there are examples beside borborygmus and tintinnabula.
As for describing these words with as short of a phrase as possible, I might tongue-in-cheek suggest reduplicated onomatopoeia or onomatopoetic reduplication. Or just call them "words that combine onomatopoeia with reduplication," though that is admittedly a mouthful. How you ultimately would do this depends a bit on context which I do not know.