first off . . . both are good guns.
I went through the same thing several years ago - instead of the LCP though, I went with a Bersa Thunder 380 CC for carry. However, I'm more of a "revolver guy" and soon got a 357 LCR. I carry OTB and I really like it. It's light yet accurate at SD ranges. I have the option of 357 or 38 Spl. but I really like the 38 Spl. cartridge and that's what I now carry in it. I can easily carry the LCR in a coat pocket if necessary. However, I'm now considering a LCP just to have for carry in a pocket holster when it's warm weather out and it would be less noticeable.
If you can, shoot both of them and then decide. I'm guessing that either way you go, you'll be very happy with your choice.
If a pair of '51 Navies were good enough for Billy Hickok, then a single Navy on my right hip is good enough for me . . . besides . . . I'm probably only half as good as he was anyways. Hiram's Rangers Badge #63