Ironically, Shadow Quagsire’s bulk goals parallel the non-Shadow’s in Great League PvP. 112.68 Def with 163 HP is the recommended minimum, enabling the Sableye 0-0 (Mud Bomb) & 1-1 (Aqua Tail), while potentiating the Gligar 2-1, Umbreon 1-0, and Powder Snow Shadow Alolan Ninetales 2-2. Even higher Def helps vs higher Atk weighted Sableye, and may also enable the Froslass 0-1. The divinely high Def of the 0/14/8 and 0/15/6 could catch Pelipper (0-0), mirror focused Froslass (114.15-114.46 Def), and possibly even Lickitung off guard, but the drop in HP could cause problems in the Sableye matchup.
Given how difficult it is to get a high bulk Shadow Quagsire, you may have to settle for a lower bulk one. 110.26 Def is a soft minimum in this regard, holding on to the Shadow Alolan Sandslash 1-2 (Mud Bomb) and Emolga 1-2 (Mud Bomb), while avoiding Carbink’s Rock Throw Breakpoint (fortifying the 0-1, 1-2, & 2-2). Coupling this lower bulk with 110 Atk could enable the Medicham 2-1 (Mud Bomb), Shadow Venusaur 1-0, Noctowl 1-1, Cofagrigus CMP, and possibly the Carbink 1-2 (both) & 0-2 (Mud Bomb).
If you really want to terrorize a Carbink, 111.89 Atk enables Shadow Quagsire to take down the Rank 1 Carbink in the 1-2 with either Aqua Tail or Mud Bomb, and in the 0-2 with Mud Bomb (112.22 Atk vs Best Buddy). With this Breakpoint, the Mud Bomb Shadow Quagsire would be 1 of ~4 (IVs depending) non-Steel-type Pokemon to subdue a Carbink that’s up 2 shields without having to spend a shield or bait shields. Neither Steel-types nor Clodsire will want to fight Shadow Quagsire either. Having high bulk is still recommended, as an Atk weighted Carbink could flip the script on you. Additionally, having >109 Def could maintain the Shadow Venusaur 1-0.