I’m also thinking of not having the Curse actually start immediately.
This.
First have exploration and adventure, with no stupidly fast countdown (it takes MONTHS to travel through Chult)...
Then, when the players are getting sated on jungle dangers... When the characters have levelled up enough... When they're getting close to finding Omu ANYWAY...
Then and ONLY then do you set the ball rolling. If you have one or more PCs affected, you don't even need the NPC or her quest! All you need are some pointers, that let them find - and reach - Omu in just a few weeks.
Think of it like this: if you think sixth level is a good starting point for the Omu chapter, you should probably spring the curse on them no earlier than eighth.
Before they complete the quest, there will likely have passed a month, even if the actual Omu and dungeon part happens in just a few days, and that's 30 hit points. Even at level 8, 9 or 10 (assuming a few level ups) that's non-trivial.
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