" Is there a fix for this? We do NOT want to replace any door trim at this time."

No, there is nothing shy of the Band-Aid you mentioned which will be messy and probably not attractive as that is a lot of wood filler.

As millworkman states, the 'fix' is new trim at the floor level. A finishing carpenter can do this for you. You could trim off the door frame and continue the wall trim in it's place. It is a very handsome look...but it is $$$.

If that is not something you want to do, then go ahead and do a paint patch on the bottom of the trim and then ignore the gap. There is nothing you can 'stuff' in there that will be appropriate. Anything you throw in there will PREVENT a nice 'fix' in the future. You will make an ugly (but functional) situation worse. How bad could it be? It could be so bad that it effects both the hardwood finish AND the wall. It could be that bad.

Just a paint touch out and then leave it alone. Anyone who is lying face down on your floor in your doorway saying "Dude! You didn't do the door right!" is probably having some other type of issue.

That doesn’t look like much of a gap. Having said that, would I be lying on my belly on the floor performing cosmetic surgery on it? Absolutely. But I’m REALLY good at it. 😬

The damaged paint is more noticable than the gap. Just repaint and have it look nice and leave it be.