Time to invoke some Clarkean Magic
I know two languages: my native English and Finnish. English tends to put the subject before the verb. Finnish tends to put the subject after the verbEnglish tends to put the subject before the verb. Finnish tends to put the subject after the verb. (thanks @JaniMiettinen for refreshing an old memory.) English prepositions are individual words. Finnish prepositions are suffixes. English can make some pretty complicated sentences...
But Finnish can make some enormously complex words...
...because Finnish has endings that convert verbs to nouns and nouns to verbs and produce scope and establish ownership and operation and a zillion other things. Frankly, Finnish can express in one word with character count X what would take English a lot of words with character count Y and X<<Y.
I'll be honest with you, without Clarkean Magic, the idea of translation from one language to an intermedite language to a third language in real time is unrealistic. It's one of the reasons why most authors either don't deal with the details or invent the proverbial babel fish. I'm going to underscore the problem with the idea of alien, truly alien, life. While human language covers a massive range of sounds and actions — we all have access to those same sounds and actions.
An alien mouth, larynx, lungs... and more... may be very, very different...
What's the point?
There comes a time in every worldbuilder's experience when one must decide whether or not the detail makes any sense. In this case, I don't think it does. It would be (and is... and will be...) difficult if not impossible to create an accurate translation without waiting until the end of at least a sentence — and frequently more. You should take some time reviewing questions over at English.se and English Language Learners.se. It's frequently impossible to understand the the question without havaing more than one sentence because context is needed to translate the sentence.
Example: "I'm sorry" has two translations in Finnish:
- "Mina olen Pahoillani"
- "Aanteksi"
Without knowing the context, the English phrase can't be accurately translated into Finnish. Frankly, without context it would be hard to translate either of those Finnish phrases into English.
But let's invoke some clarkean magic
To a degree, your "system" can look ahead and predict what the probabilities are of what will be said next.
It can do that with reasonable accuracy because it's looking behind and keeping track of the conversation, so it understands the context behind what's being said.
And it does all this for every language known at such a high speed that if it happens to detect an error in translation, the correction is taken in stride as we do when we mishear or misunderstand someone.
In other words, with a bit of Clarkean Magic, you have a solution that wouldn't be disimilar to pretty much any conversation humans have with one another right now.
But to try and work out the details of how this happens? IMO, that's a waste of time as any solution you come up with will likely be inadequate even for human languages, much less alien languages.