You have various choices. Assuming these are props, you can parent every part to to one part, then moving that one part will move all the other parts (If you want you can lock all the parts except for that one main part). You can then use 'Select Subset' when saving to save all the parts to one pp2 (prop) file.

You can load a figure that has only one part, the BODY actor, parent all the parts to the BODY, then save the figure back  to the Figures palette. Such a BODY only is "MinFig", available from my free stuff.

You can export all the parts as one obj file. Seems like you already tried that, but it did not work as desired. The default Export Options settings will not preserve the grouping. Try exporting using only "Include existing groups in polygon groups" (see attached image).