I have a D700 and D2Hs. The D2Hs gets it's most use with my grandchildren. It's a huge camera, tough to use at times because of it's size, but with the 17-55mm f/2.8 AFS DX lens, it's a near perfect range for use indoors with kids. The sensor is not as sensitive as your D300, but still does quite well. I've learned to use my SB-800 with the camera, using it as a flagged bounce strobe. That gives excellent modeling on faces and if done properly doesn't necessarily look like I've used strobe.
The D700 is equally as effective, but I like the D2Hs especially with my 50mm f/1.4 AFD or 70-200mm f/2.8 AFS lens. The extra battery power and excellent AF module makes if a very quick to focus and responsive camera to use with fast-moving children. The vertical grip is a strong asset for portrait mode and aids in managing the 70-200mm lens.
I also use a LX3 and X100. I like these cameras more for travel when the subjects are not moving so fast. Both the X100 and Lumix LX3 are very capable cameras and with skill can be used very effectively with kids but I've found that they are not an exclusive solution to family and children photography. Mine get used most with kids outdoors. There they do make sense where the field of view is a bit wider, lighting levels higher, rapid lens response not quite so necessary because of DOF, mostly on family outings - e.g. like in travel photography.
However for those all the time precious moments, the DSLR with favorite lens and even a strobe (when used properly) is always ready on some nearby shelf when the kids are around.
When my D2Hs gets to feeling too large, I mount a 50mm f/1.4 AFD. It's still a big rig, but your D300 would be just a capable, you might prefer a 35mm f/1.8 AFS or 35mm f/2 AFD for FOV - that's a matter of personal preference, both the 35 and 50 are very effective indoors. When I use my D700 in that mode, I'll often mount my 35-70mm f/2.8 AF. It's a "smallish" zoom that works well with that camera indoors.
So the short story for me is make maximum use of the DSLR, it's a "can't fail" kind of camera. You may find the others you are considering, while satisfying the size requirement, also have some photographic drawbacks.