I had a dog once that got a hold of a disposable contact lens in the wastebasket. He ate it and died a gruesome death. Without my contact lens, I couldn’t tell for myself, but I was told it was not a pretty sight.
If it’s a plastic bag, it would be best if it were disposed of it in the trash. Ideally, the bag would first be emptied into a toilet, but I can see how this might not be feasible for you.
I gave a more complete answer in the related tampon thread, along with a link to one municipality’s detailed list of items not to dispose of in sinks and toilets, as well as a list of “Do’s” and “Don’ts.”
While I see that these bags are marketed as “flushable” in the link you provided, there is no recognized standard for what “flushable” really means. All I can tell you is that the sanitary sewers are only designed to handle wastewater, human waste, and toilet paper. (In that other thread, in this post, a poster linked to some recent problems the growing popularity of so-called “flushable” wipes have been causing.) The reason why plastic bags are problematic is because if they are sealed and stay intact, they can easily block a pipe or a pump. If they shred, they can wrap around a pump impeller.