Enjoy the time with your children too. The memories you build with them today will last a lifetime, and will be much more valuable than any side project you're dreaming of
But I'm blessed that my kids are sleep through 99% of the night time.
...or do your children not wake up at 6am sharp every day? :P
But what I say is - I'm not doing that 7days per week - if I'm tired I watch some Netflix or just play games or just go to sleep (but I feel really guilty that I've wasted free time to go 'just to sleep' - I know that's a bad mindset, I'm trying to fight with it, because sometimes your body needs to recover a bit more)
I got to a point that I couldn't bring myself to even open a computer after I got home from work ... and while I recovered to an extent (I've had a number of successful (IMO) side projects, and even did some startups since), I haven't quite felt that drive to spend my off-time doing more work. My daughter just turned 18 yesterday, and I'm looking forward to when my son is off to college in ~4 years; side-projects, here I come!
Took a long time to realize, and of course YMMV. I had kids younger than most of my peers (~10 years early, when I was 20 ... it was all planned, perfectly on schedule), so when a few of my friends' wives became pregnant recently, I gave them all the same advice -- to give yourself time and space to learn how to integrate this new aspect into your life. I went into it trying to maintain my "me time", with side-projects, writing books, making games, reading/writing blogs, participating in online communities (/. at the time), etc ... and I burned out, so my advice to them (and to you) is to learn how to re-balance your priorities. Beer and TV will kick your ass the next day if you stay up late, and eventually it'll catch up to you. Better to go in cycles ... have some "benders" from time to time where you hang out with friends, play games, drink beer, watch TV; and then have a stretch of time that you go to bed at a reasonable (and consistent) hour, exercise, etc.