I would suggest Beddit.
The things that it has going for it is that, since it detects your presence through the strip affixed to your mattress, you can get up repeatedly, and it will track when you are actually asleep. One thing that is useful is that each time you get up and get back into bed, it will show how long it takes to fall back to sleep.
It gives you a sleep score that adds up all time slept during the day, including naps. It shows the time that you sleep in a nice timeline format that I think would work with your situation.
In addition to time slept, it tracks respiration rate and heart rate, which I find to be interested. With both of those, it will tell you if last night's metric was more or less than "usual". While vague, I find it to be interesting feedback.
The cons for Beddit include that fact that it doesn't have in-depth data on your sleep phases. It has a visualization that shows when you "dip" into deep sleep but that's it. There is no metric for how long you've been in deep sleep.
Your phone's app (android and iPhone) has to be open and actively measuring sleep. Even though it can still measure in the background, if you use your phone while doing parenting duties, this could introduce complications (since the app is seems more likely to crash when it is running in the background).
Another con would be that Beddit has no meaningful data export, but since Beddit connects to Apple Healthkit, you can use the QS Data Access app to get your sleep data out.
This is based on my personal experience. Another tool that is similar that you may want to look into is Withings Aura.