"They will have been happy together" doesn't make any sense in English. It is mixing future and past with no mood context. It implies that you looked in the future and could see that they had been happy in the past (which is your future). If this is a tense that cannot be translated, then it should be translated directly as a lesson.
The translation of -
'Ils auront été heureux ensemble' = 'They will have been happy together'
is correct but I know it does sound strange without further context.
Take a look at the answers to similar comments in the Q&A below that lesson on the futur antérieur (future perfect) given by Chris and myself as they should help you understand the concept.
Le futur antérieur describes a future action that will have happened by a certain time in the future.
Hope this helps!
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