Heh, I wish I’d met you and this board about, oh, 7 1/2 years ago 
Three kids later and ahem much older, I’m well-experienced with the sleep deprivation and such that goes with small children (our youngest is 3 this month). I’m just starting on the lucid dreaming thing and while I’ve yet to have one, I feel close every so often. It’ll happen. I just need to be diligent.
Something about sleep deprivation that I’ve read, that I’m trying to put in its right place is that it causes you to go into REM more quickly. By that logic, you should be in dream state more quickly too. Other things I’ve read have said that you need to get a good night’s sleep in order to get to the later periods of REM sleep, which are longer than those earlier in the night (or whenever you sleep), the rationale being that the longer you’re dreaming, the higher the chance that you’ll realize your dreaming and so on.
I realize this is drifting a bit off-topic, so I’ll bring it around. I’m not sure what effect the longer term sleep interruptions will have on dream patterns. I know that with our first child, the first few weeks were exciting and exhausting. By six months, much of the thrill was gone, and I just desparately wanted to sleep whenever I got the chance. Things stabilized somewhat and then we had our second and it started all over again. Our third was a better sleeper than the first two (who had acid reflux which could affect their sleep), but by this point I had a bit of a sleep debt built up 
It’s entirely possible that learning to LD from this position would be harder than keeping it up. I have just enough neurophysiology in my background to be able to fake up an explanation in either direction. Keep us posted, though. I’d be curious to hear what your experience is.
I split the off topic discussion about raising children into a new topic here