This implies that the treasure you trade for must be loaded. What happens if you receive a UT that can’t be loaded, such as Wolves? I assume it would apply to any treasure received via the swap, and not to the treasures you took originally and didn’t swap? (because it occurs after exploring, and after step 8 – “Apply the effects of all revealed unique treasures.”)
The UTs are treated as if they were on the island before you did any swapping. It does mess with the timing a bit, but in the case of something like Wolves, that could force you to put back others that you technically just loaded.