That makes absolutely no sense at all. The numbers 24W@200F and turning it up makes it burn, tells me that there is really something wrong with your build/wicking.
These are the Notch coils I use.
$4.37 Demon Killer 316L Stainless Steel Notch Coil Resistance Wire RBA Atomizers (10-Pack) - 0.2ohm / quick heating at FastTech - Worldwide Free Shipping
I have used them on the Pico, Cuboid and DNAs, in TC and working perfectly. The 5mm coils will take a full strand of Rayon to fill it properly and don't really come alive below 35W. I run them at 50W/480F on all my devices for a very dense and flavorful vape. These notch coils are not really designed as a tootle puffer coil and don't perform well under low power IMHO.
I have never seen anyone with so many obscure problems...

Too bad that you only have the Pico and Merlin to use. It would be interesting to see if different gear would cause you as much trouble.
Something like this possibly...
Innokin iSub V Cool Fire 4 TC100W Aethon Chip Starter Kit