I recently bought a Bodyguard 380. I have never shot the LCP, but I've owned and carried a Kel Tec P3AT for two years. Since it appears to be identical to the LCP, I decided to go ahead and vote.
I vote for the Bodyguard 380, especially if you're going to get a laser for the LCP anyway.
To me, the Bodyguard 380 is the gun Ruger should have designed and built, instead of copying the Kel Tec P3AT and charging more for it.
The Bodyguard has better ergonomics and an overall better design, imo. For those that must have a safety or they don't feel comfortable carrying it, it has one. For those that don't feel comfortable carrying a gun with a safety, the trigger pull is strong enough that you don't need to use the safety. The sights on the Bodyguard are much better, and I found that the gun is very accurate without the laser.
As to how comfortable it is to shoot, to me both guns are uncomfortable with very long trigger pulls. I was actually surprised that the Bodyguard's trigger had such a strong pull. I guess, with the safety, I figured the trigger would be a lot easier than the Kel Tec. Well, it ain't. The Bodyguard 380 has a strong trigger pull and it took me about a clip and a half to get used to it. Once I got used to it, I found the gun to be very accurate. And, although I like the ergonomics of the Bodyguard better than my Kel Tec, both guns are small, light, and will jump when the trigger is pulled. It's a gun you really need to practice with, imo.
As to reliability, the Bodyguard 380 had zero problems through the first 100 rounds. My son and I shot it fast, slow, one-handed, two-handed, etc. -- no jams. I was very pleased with it. I found my Kel Tec to be reliable as well.
What sets the Bodyguard above the Kel Tec and LCP, in addition to the ergonomics and better sights, is the integrated laser - it's nice to have the laser option when/if you need it, and I just like the fact that it is integrated into the gun itself, as opposed to an add on. Plus, the LCP may actually end up being more expensive than the Bodyguard once you add a Crimson Trace laser to it.
So, I like both guns and I am not giving away or selling my Kel Tec. But I do like the Bodyguard better and I am carrying it right now.