Your proposal is a good sentence but with a different meaning. He must have been bringing ... is a little closer. “He would have to have been ...” expresses why you think what you do. It follows from the logic of the situation. I concluded that he was bringing .... I concluded from the evidence that he was bringing ... I deduced from the evidence that he was bringing ... The evidence indicates that he was bringing ... Would be similar but not better. He must have been bringing ... is also similar but not quite as definitive.

'Must have been bringing' would work - it's a bit shorter and it keeps the continuous aspect. And note that in spoken English, this wouldn't seem especially cumbersome. The modal + auxiliary + auxiliary combination, 'must've been' would trip off the tongue in a second or so as something like "must'u'bin" - no problem at all.