You state your LCP shoots hopelessly low and to the right.
Your shooting fundamentals are not being put into use and what is happening is predictable.
When you watch the laser dot on the target you are putting all of your efforts on holding the dot right there and if you did that with the sights on the gun the results will be much the same.
Yes laser sights are very good but one needs to be good with the gun sights also.
I also have an LCP and am extremely impressed with the accuracy.
I put my first laser sight on a Ruger single six and used it for shooting rats and when not at the dump (back then) I would shoot 12 gauge hulls. When shooting off hand I didn't do that well but if I anchored the pistol at my hip I missed almost nothing. Yes they are fun and good.
For the message of the Cross is foolishness to those who are perishing, But to us who are being saved, It Is The Power Of God. 1Corinthians 1-18