If you want to carry on an empty chamber, I say go for it. However, do so in a holster that you would consider using when carrying on a loaded chamber as well.
What I recommend for people who have the same concerns as you is this: carry the gun without a round in the chamber, but keep the striker cocked. When the day is over and you go to remove the gun from the holster and put it wherever you do at night remove the magazine and check to see if the striker has fallen. This will show you that the gun would not have discharged while you were out and about during the day. I tell them to continue to do this as it builds their trust in the safety features of the gun and the holster. This also works for those who want to carry a 1911, but are hesitant to carry cocked and locked.
I say give this method a shot (no pun intended) with your LCP, it will probably build some confidence in your carry system, and before you know it you will be carrying with a round chambered.