I would rule that you absolutely can use the second half of Ignite to perform a final function before deresolution, even when you are doing it to yourself through Conduit. The two special abilities were written this way to hint at this interaction. It's perhaps debatable just how much extra "story" impact a PC can throw in with a normal function roll in their last breath after they've already just been responsible for one of the World-bending Authorities, but I'm sure that maximizing this interaction will be a fun challenge for somebody.
However (rule as written) I would want that function roll to be one of the 9 ordinary actions on the player sheet. Conduit could maybe be written a bit better to reflect this, but the end-of-scene Deresolution is intended to be the cost paid to access the ability (even though you pay the price afterward). You've got me thinking though about a possible rewrite that does permit the player to just reflexively Authority when they are on their way out of the door... or at least clarifies this interaction a bit so that Conduit has a more clear opportunity cost.