The first 5 or so years I rode, my instructors were always telling me to keep my heels down. When I didn't, my foot would either go through, or I'd drop it.
So I never really got used to keeping heels down, except when I knew the horse would spook.
To start off with, maybe raise your stirrups a bit. When they're too long, you can't put heels down.
I learned Balanced, or Centered riding. In that, you learn that straight back/shoulders far back, isn't needed, and that you can ride easier, and balance better, if your heels aren't down.
However, you need to make sure that you have your feet right...
Just sit on the horse for a bit, and try this.
Stick your feet in the stirrups.
Now, turn your foot until it feels like you're putting the weight on your little toe. If you have a mirror on the wall in the arena, if you look, it will look like your foot is level, and you'll only see the side of your boot.
When you put your heels down, you normally stick your toes out, and have your heels closest to the horse.
If you shift your foot so your weight is more on your little toe, and not your big toe, then you can totally relax and still have your foot in the same position.
It took me a few days, but I eventually got it.
If you/your instructor still want to keep the heels down, it's not so much bringing your toes up to force your heel down, but it's putting all your weight in your heels and stretching your calves as far as you can.
My instructor would come over, take my heel, and force it down, which somehow made me put weight in my heels and stretch my legs.
So, after my long post, try to put all your weight in your heels. No forcing toes up; just drop everything into your heels.