I mentioned sheeting out over the entire floor because I thought you were 3/8" in 8' ( over tolerance ) . But 3/16 is barley within.
You could just fill in the "dip with 1/2" plywood. It would be a little tricky without sheeting out the entire floor, and over the leveled dip.
Screws would be best for fastening. But here's the tricky part. You would have to cut your plywood shape, and do your sanding before securing it to the floor. Do your cutting, sanding , or planing, until you have the perfect insert for the dip. I would temporarly tack the plywood down, in the middle , to achieve the flatness, and feathered edges that you need. It will definetely take some patience. Obviously you cant screw , or even countersink nails , and then be able to sand, or plane a feathered edge.
( you would be hitting screws, or nail heads).
Filling would sure be easier. But there's no guarantee a filler wouldnt loosen in time, with movement, over that Fir subfloor. I imagine you might be able to fill if you screwed down lath first. I just havent tried that process. And then there's the feathered edge to deal with.
Tile instead ? After all this?
Just kidding
I'm not a tile contractor . But I believe theres a couple options. An Installer would lath over the entire subfloor, and float a morter bed. Or fill the dip with thinset, and thinset , and screw backer board over the entire subfloor.