Yes, we do use W3 total cache. We have object caching always disabled since this slows down the admin quite a lot.
Switching of the cache completely is not an option since speed is then back to zero.
We also use the combination of PDF invoice + W3 cache in another store, no issues there.
The main difference between those stores is the use of the “WooCommerce Order Status & Actions Manager” plugin.
For me it looks like an issue connecting the PDF invoice to the primary order status. This is filled in correctly, but does not appear to work.