Courage should have this in the bag.
The biggest thing preventing Courage from just directly leaving with the Bagges is the Spiral Curse's Spatial Manipulation causing them to be incapable of conventionally leaving the town, but Courage has methods like his Toon Force, Fourth Wall Breaking or Immersion to circumvent them (such as leaving through a painting or a cyberspace), and the Magic Tree is kind of an instant-win given that Courage could simply just wish himself and the Bagges out of the town. Computer is also a major help to Courage here, since not only does Computer tend to know a ton about magical or archaic things within the show (though it's not all-knowing), Computer could also be given something like a spiral drawing or a mutated citizen in its scanner and figure out what exactly is happening. Courage himself is also very smart, way smarter than Kirie (the protag of Uzumaki), who was also semi-succsessfully capable of figuring out the mechanics of the curse even if she couldn't escape it.
The Spiral Curse also can't actually put Courage down since Courage can resist its biggest hax like Mind and Disease manip, and he has good regen The Spiral Curse can't circumvent, so trying to physically kill or overpower Courage won't work. Eustace and Muriel could end up becoming transformed by the curse, but there are multiple examples of Courage in the show still capable of saving the Bagges even when they were heavily diseased or transmogrified, so there'd be nothing stopping him from finding out a solution either within the town or after they leave. The Spiral Curse's Time Manip also won't help it here given that Courage and the Bagges wouldn't be subject to any aging effects, while the Causality Manip would only keep throwing natural disasters Courage's way which he can also avoid with his stats. As far as stats go, while the curse could make bigger natural disasters to try and trap Courage, Courage with a scream could just blow it away, and as a last resort there's nothing stopping Courage from just screaming so loud he ends up obliterating the whole town. If this did happen and Courage ends up uncovering the previously built towns underneath, maybe he manages to also save the trapped citizens too take that Junji Ito.
As far as the victory conditions:
Figuring out the curse and its mechanics in under 48 hours? Easy dub for Courage
Saving Eustace and Muriel? Easy dub for Courage
Actually escaping the town? A lot harder given that he could end up being swarmed by the citizens, but Courage even if he was cut off from his equipment still has better stats and his own natural powerset that'd let him physically take the Bagges with him away.
Courage manages to win the day despite the horrors once again.
