Lightroom: can you export the ‘picked’ flag to another software? Alternatively, can you add a keyword based on the picked flag?

Ideally you would write the Lightroom metadata to sidecar XMP files and use those to import all metadata into another non-Adobe application.
Unfortunately pick/reject flags are one of the those things that are stored in the catalog only and are not written to XMP1,2.

Therefore your proposed solution of using keyword seems to be the best option.

To do so, you need to first find all your images with a picked flag:

  1. Go to the Library module
  2. In the top, click on Attribute in the Library Filter:

enter image description here
3. Press the white flag to show only picked photos.

Now you need to assign a keyword of your choice to all these photos:

  1. In the Library module select all photos that are shown while the Library Filter set in the previous steps is active
  2. In the Keywording panel on the right add the keyword of your choice (I chose picked in this example):

enter image description here

  1. While the photos are still selected, make sure that the metadata (i.e. the keyword) is written to the (sidecar XMP) files by going to the menu item MetadataSave Metadata to Files