I have been stuck on the same daily, getting the 12 pieces of flesh, for 4 consecutive days and I am assuming today will be 5. I have done all 3 of the pet battles along the questline so what is up?
Update: Fifth day in a row now the More Fetid Filets is my quest.
What is your progress on the drone (the yellow bar over its head)? It can be very hard to read the number, for some reason it continues to shrink down even when you zoom right in.
You may already be up to the next stage where you talk to the old Uldum quartermaster in the north east corner of the compound, and the drone trainer is just giving you more dailies because you haven’t moved on yet.
The progress bar has gone away, no I am not having a UI issue because I see other players progress bars for their drones still. And yeah I have went over to that trainer to talk to him just in case but nothing there yet either. I believe I “should” be at that stage but it isn’t coming up yet.
Update: Day 6 of the More Fetid Filets daily hostage situation. Not really feeling like anyone at Blizzard is reading these bug reports any more 
From my experience, I feel the dailies such as Fetid Filets and Honing Instincts are simply filler until the Black Empire, or minor assaults swap locations, as some of the quests seem to require a faction in action.
I have had the quests alternating all the way up until it looked like my pet was going to hit the next level to get to the end quest but yet has been stuck on this same one for almost a week now. Still waiting on someone other than a GM, who seemingly can’t help at all, to respond to this from Blizzard.
Update again: Had a gm respond a few days ago and said it looked like I needed to do 3 more days of dailies to level the pet to the end. That time has come and gone and today I have no daily to do nor does the next guy in the questline have a quest for me. Will someone at Blizzard please help me get to the end of the questline?!
Another update of sorts I guess, still no new daily or quest to keep progressing going on 3 days now. Blue post respond please?