I think in theory, yes he should. Just being a SAHM in and of itself does not mean you should get no sleep
In practice its more complex, of course.
For us it actually comes down to who has to drive that day. I really don't think its a good idea to drive while you are knackered.
I actually did do pretty much all of the night time stuff the first two times, simply because I didn't HAVE to drive (we live near where I worked) and dp actually can't really get to work unless he drives. It was my choice for him not to do the night stuff, as I REALLY didn't want him driving when knackered.
Oh and saying all that, we did actually sleep in shifts to an extent-I would go to bed at 7 and sleep til 12, while dp drove ds round and round and round. Then dp slept til 7 while I had him. And at weekends, he DID do nights.
This time round, if we get another crap sleeper (and my god we breed them chez filly ), it will be more difficult, as I do have days when I NEED to drive.
When I went back to work, incidentally, I still did the weekday night stuff, because of this driving thing. IME, there isn't much in it between working on no sleep and looking after kids on no sleep IMO. Both are hell. And while I wasn't a teacher, I did have the kind of job that required presentations, court appearances, etc, so its not like I could slink in and hide behind the coffee machine all day.