They are both classic instruments, each with their own signature tones.
If the producer has asked specifically for a Jazz, it might work better for the vision they have in their head and are trying to materialize.
Generally, it is usually good to follow the Producer's advice, there is a reason, after all, in their existence and in the band choosing them.
I don't think, though, that bringing more than one instrument to a recording session has ever hurt somebody.
Bring both the Jazz and the MM and choose in accordance with the band, the producer and the engineer(s) what is most fitting for each track.
If the producer has asked specifically for a Jazz, it might work better for the vision they have in their head and are trying to materialize.
Generally, it is usually good to follow the Producer's advice, there is a reason, after all, in their existence and in the band choosing them.
I don't think, though, that bringing more than one instrument to a recording session has ever hurt somebody.
Bring both the Jazz and the MM and choose in accordance with the band, the producer and the engineer(s) what is most fitting for each track.