It would seem that you need to persuade paperpile to export your reference list as an xml file and then look to see if that can be imported into Word. This would be your better long term solution if Paperpile can't provide an appropriate plug-in or export filter.

You could ease you pain a little by avoiding using list numbering and working directly with bookmarks.

Consider the following alternative.

Create a new caption type called reference

1. References.Insert Caption
2. Click on new label
3. Type 'Reference'
4. OK
5. Check the 'exclude Label from caption'
6. OK

You can now insert your references using

References.Insert Caption

You can now insert cross references using
References.Cross reference then scroll down to select References.

This means you will be less prone to errors when moving references.

The downside is that you will have your references in Caption style but you can always apply a character style or local effect to change the appearance.