First off, never, EVER, turn on your heel. For some surfaces, notably, foam padding, one gets the feeling that the turn is much easier and more controlled. And they would be right.
But then move on to finished wood or carpet, and the surfaces can get more slippery. Also, foot injuries are more common with heel spins than with ball of toe.
Finally, you lose a few inches of your maximal range when you use heel instead of ball of toe.
That said, how to prevent grinding your foot into the floor? Your weight is all on that 2 inch surface area, and if you are grinding the ball into the floor, and your ankle or knee will absorb the torque and become injurious.
For most turning kicks, esp for heavier people, I explain that it is possible to slightly elevate the body - not quite jumping, but close to it. In this manner, less weight is taken off of the ball of foot. Yes, this creates a slight pause in the technique which could be picked up by an observant sparring opponent which makes this a telegraph. But that is a sacrifice you should decide if it is worth to you.
For some, a kick that causes difficulty is a spin kick. For me, I turn my front foot backward 180 degrees away from my target, then spin with the rest of the kick - but my foot doesn't move until well after kick delivery and my weight can safely shift. Over the years, I've developed optimal ankle movement that allows me to do this, but not everyone can do this right away. In this manner, I have no torque applied to the ankle or knee. (you didn't ask, but I pay a price when I execute other techniques, such as the low spin kick; so this can be a double-edged sword)
But you mention specifically the 360 kick; there is a trick you can employ here.
If the 360 RH is done properly (and improper execution could be your problem), you should be turning backwards so that your rear knee spins up and toward 10:00 (for a left foot kick) (and 2:00 for the other). This lift has the effect of reducing the weight on the standing foot (ball of foot), and so should reduce the chance for injury.
If your turn is too slow, then you will look like a dancing bear trying to balance on a ball - very uncoordinated and slow as you take a few hops to get around.
If your turn is fast or slow, but you fail to kick up your knee for the 10 or 2 o'clock position, then you are only grinding your foot into the floor which will cause the fatigue and injury.
So to get this lift, you need to spin backwards fast such that the knee will move upwards; this allows the momentum to carry you in an upward spin, thereby lessening the weight. Indeed, you should be airborn halfway before you reach the target. Many beginners can't do that so soon, as they get airborn just before the target and so they spend the entire kick grinding the foot into the floor.