Unfortunately this is an inherent difficulty with how WordPress is structured (and mainly its DB schema – which is not that optimal). If you also need to keep ID references exactly the same the situation gets incrementally more difficult because you need to make sure that already assigned IDs are not occupied in the new installation.
I can’t see any other way other than:
1. Try to export the part of the database where playlists are defined (with a really complicated query – which will also have to contain image files) and somehow import them back to the new database. I don’t see this as feasible.
2. Export *everything* on your old WP installation, including images, playlists etc, either via the WP import/export plugin or with a DB migration. Then start actually bulk deleting the stuff you don’t need (e.g. everything except WooCommerce and AudioIgniter playlists).
I think if I had to do this I’d definitely go with option no 2.