There is nothing in the structure of the verses of the song of Moses (such as clear counterpoint) to suggest a two-part duet or even call-response.
There is no indication of duet in the OT, only of call-and-response and no tradition of duet in ancient post biblical pre-Christian Jewish text and no continuous tradition of duet preserved in any Jewish community.
The commandment in Deut 31:19 is to Moses only. The verb forms in 31:30, 32:44-45 are all singular.
The verb used in relation to this song is דב"ר to tell or say, not שי"ר to sing as in Exodus 15:1, 21, Judges 5. So there is no clear indication that there was any singing going on at all, only teaching and telling.
In Exodus 15:1 and Judges 5:1 the participation of others in the song along with the principal singer is mentioned before the song. Here only afterwards and obliquely, and framed by an additional verse, 32:45, mentioning only Moses.
There is no indication in the Deut 31 where the commandment is given that Joshua should take part in teaching this song, he is only mentioned afterwards in passing (and that is the real question here).
The parenthetical mention of Joshua at the end of 32:34 can be explained as referring to the the first verb of the verse, Moses came, (i.e. Joshua came with Moses) in support of the narrative of Moses's death that follows and the mention of Joshua in 34:9.
There is no indication in the 31:19 of how the song should be recited, only that the descendants of Israel should be able to recite it from memory. So the speculation in this question is not well founded.
