General question: could the be such a thing as a slow Imperius, so that the effect looks to everyone like a natural change of mind?

Sure, why not? It's magic. A slow Imperius is no weirder than many other already used spells.

What does canon think? Well... I believe our only real POV from an Imperius'd person is when 'Madeye Moody' casts Imperius on Harry during a Defense class, and IIRC Harry's mind is instantly & profoundly messed up even before Moody issues any commands; likewise, the imprisoned Barty Crouch is well aware that he is being controlled. Similarly for the Gringotts Goblin in Deathly Hallows.

The chapter title, IMO, points to clever social engineering, not magic.