I agree with Pyrros' suggestion, I would attempt that approach first.
You mentioned that your horse doesn't respect/ is not sensitive to the whip. If the previous method has no effect, I recommend that the next time you lunge the horse and they step in, step towards them aggressively. A horse that bows into the circle on the lunge is invading your space and if you make it clear that it is your space and they will be warned and punished as they would be in a herd in the same situation, then the horse will respond and eventually stop testing the boundary of your personal bubble. Especially if the horse does it consistently in one place, you can plan in advance so that the moment they try to make their move, you're already confronting and redirecting them.
When you step in aggressively - make it a sharp, confident movement. you can move the whip or snap it, make yourself look bigger. And keep going. If you back off, then they will come to understand they can invade your space at leisure because you'll yield to them.