Taking toddlers photos are more difficult than most people think. They move fast from place to place and do not have patience to pose for anyone. You literally have a split second to capture the moment. I know because I have a very active toddler and that is the reason I got my D-SLR system. The most difficult ones are low light and indoor photos. Shutter lags are the worst when light is dim. But since you don't want to carry D-SLR, finding a compact camera that can have decent performance in low light is going to be challenging.
The Panasonic Lumix DMC-LX3 has a very fast lens f/2.0 which can be a great help in taking low light photos. But it does not zoom out much, which may not be a problem since you are usually within a short distance from your toddler anyways. You should learn at least to use the semi-auto modes like aperture-priority, it's easy and will help you to take better toddler photos.