I used to keep my Baby Browning .25 auto in my seat bag, and later a Seecamp .32. I think you could do the same thing with an LCP. If you're attacked by a dog, you always keep the bike between you and the dog anyway, so I didn't see having the gun stored there as a big drawback. (I didn't plan to shoot anyone's dog anyway.) I rarely got off the bike during a ride, so I was not concerned about someone stealing the bike and thus getting my gun, or needing to take the gun inside with me somewhere.
There are also frame tube bags that you could use to store a small pistol. Most fit between the top tube and the seat tube, and are roughly triangular in shape. You might consider one of those.
A really lightweight gun like a P3AT could be carried in your jersey pocket, but I never wanted to bother with that. There is the sweat issue, and if you fall, you may lose the gun, or get it badly dinged up. I always thought it was better to have the gun in my seat bag. I have a compact bag sold by Specialized dealers that is pretty quick to get into (at least quicker than others I have seen), so that was always good enough for me.
