Everyone wants to use PUP as the pet side, but I think PUP really shines in Omen as the DD. 3-man Omen bosses(barring Kei) aren't much of an issue for PUP as main master DD. Malignance turns PUP master into a great DD+tank.
Gin: Pretty standard fight, maybe use healer automaton to help with the AoE.
Fu: Send in the automaton for the phases of absorption, use PUP dd outside of that when it's safe, this one is probably the best for straight Puppet kill though.
Kin: Now we get into gimmicks where PUP shines. Target is shut down by deactivate-activate-Ventriloquy, which gives PUP master free reign to solo DD with a couple of support jobs.
Kyou: Again, the gimmick can be managed with Ventriloquy. Just DD to your heart's content until it gets close to using Bravado, then ventriloquy all your hate onto the Puppet. Malignance shuts down the added effects mostly, but would recommend using healer or RDM puppet for the added help along with two supports or a healer and BRD. Only bad part is juggling meteor with Bravado at 10%. You'll only want to trigger Bravado right after a weak or well resisted Meteor. Leveraging GEO Fade will help, Bolster is recommended around sub 25% due to Meteor. You'll also want to make sure everything is good MP-wise at 26% before pushing into Meteor spam.
Kei?: Maybe a hybrid master+pet set with dd puppet to help with Skillchain frequency. GEO/COR and BRD should be enough support as Kei doesn't hit very hard. The key is just keeping skillchains constant and of course out of ranging Fuller. If doing master dd, you'll basically want to try and 3-step as much as possible with VS/Shijin-Shijin/VS-VS. If you can't reliably 3 step by yourself, with slow on then I wouldn't bother. Having SAM roll, multiple marches, while having enough attack and accuracy to deal decent damage with the skillchains is the key. It needs to be consistent or you will risk timeout from benediction or regen effect.
I can speak to personally having done all but Kei this way as PUP, BRD, GEO, (tribox) and trusts. BRD might be the key piece though, it's a 4 song Carn BRD with Honor. But obviously a player is going to be more helpful getting Elegy and Nocturne back on than my alt.
Midbosses aren't too bad either. Craver just benefits a lot from a GEO using BoG/EA Wilt, standard fight once that's up. As PUP master DD you have lots more control over Pain Sync than with Puppet as the primary DD. And on Gorger, you can leverage the puppet tanking via Ventriloquy if you pull hate.