Sorry Barb. "IT" happens, so we just do the best we can with it when it hits the floor.
Unfortunately I don't have a plan that allows a fix. But I may have a plan that will allow a fudge factor? It will all come down to cost, but I have helped a few home owners redo base boards so they had a wider. and taller one that would cover out to 2 1/4" to hide flooring mishaps. Starts with a taller base in the back, must be 3/4" wide. In front of that another shorter 3/4" piece, and then a 3/4 shoe mold in front of that. If the cost comes in less than the replacement cost of new material, you still win. Saved putting down the floor, and got a new look on the base.
Plenty of variations of the "look" but most styles of decorating, and home can work with some variation. Key is to look at thicker trims, many due to costs are becoming 5/8" wide, instead of 3/4"
I think trying to glue it may not end well. Especially if the floor goes into another room where you would walk through the skinny piece. Without any traffic on it, maybe it would work? Only way to find out is to try it. You could go that route, if it doesn't work, you could change up the trim later.
However it ends, you are NOT the first person who figured the wrong lay on a flooring install.
Ohhhh, in case you do end up widening the trim, under the gap, lay down a piece of the new flooring so everything sits on a level field. Otherwise trying to hang the new trim will be a pickle. Just don't make it tight to the field, or it will mess with expansion, allow around 1/4" total.
Ok, now rereading, and seeing that piece where it looks like it will be a transition piece maybe? The trim thing will just work on a walled setting. For room to room, where 2 floorings meet, yep you need a transition. Just make it a custom one, that is a tad wider. You may have to make it thicker so if walked on it won't crack. But I know some flooring guys who have some pretty wide transition pieces, just taper it down on both sides to keep it from being a tripping point.
