Altium's Pick and Place exporter uses Comment field, to distinguish components, Is there any possibility to replace it with other fields? for example Part Number.
I believe, in the assembly house they use Designator as the main identificator. That is why we also provide BOM with Designators. But to answer your question, you could probably fill out Comment with Manufacturer PN in your library (I am not sure if you would like to do this) or you could try to adjust the Pick & Place file to add one more column with Manufacturer PN (again, I am not sure if this is possible) or you can combine BOM and Pick and place (may be an additional of work) ...
However, what could work, you could make a custom Template and generate a BOM (the standard way) and include position information about the component. For this, you have to go into your PCB: Reports -> Bill of Materials and then you will be able to select position of the component (we use manufacturer part number as libref, if you do it different way, you may not be able to do it, as manufacturer PN doesn't seems to be available in the PCB BOM):