My son is 4.
He still naps everyday... in the afternoon.
He acts like your son... if he does not nap.
My son cannot last the whole day, without a nap.
We have a routine, everyday for naps, and he knows it like autopilot.
For me, my son 'usually' naps RIGHT after we get home from picking up my daughter at school. So... about 3:00.
Then he naps for about 2 hours.
He wakes from his nap, and STILL goes to bed just fine about 8:30.
The thing is... yes it is busy after school... BUT you need to get your son to nap... so that must be a priority. IF the other afterschool stuff gets in the way... then your son will not get to nap.
So you need to decide... which is more important... to you and the best for your son.
For me, after-school when I put my son to nap... my 8 year old daughter KNOWS full well what the routine is..... while I am tending to my son, she handles herself eating a snack I made for her then the does her homework. After my son falls asleep... then I am back to her and the other routines... of the household.
ANOTHER option may be... can you get him to nap BEFORE... you pick up your stepson from school????
... my idea would be: give your son LOTS of runaround time in the MORNINGS. THEN have lunch earlier. Then, instead of having ANOTHER play time & story time after lunch while you do dishes (because that wastes time), use THAT time to WIND-down your son for his nap. (kids need to wind-down first). THEN do the dishes after or later. So you don't waste time, getting him to wind-down and nap. A child cannot nap.... IF there is horse-play or lots of hyper things before hand. Before naps... should be MELLOW boring calm things... to wind them down and transition them, to nap. For me, I even make the rooms dark... and then my son changes into jammies.
My son, it takes me about 1/2 to 1 hour to wind him down, PRIOR to nap. IT is a process.... so keep that in mind.
The other alternative is... you NEED to get him to nap, after your stepson comes home.
ALSO your Husband cannot expect... ANY child, to be 100% in a good mood all the time when he is home. That is not natural. But yes, if a child is over-tired... then they do get fussy. My son was like that last night. I explained to my Husband. But... again, a husband cannot expect the child to be smiley all the time... just because he is home.
Your so called "inability" to get him to nap... is NOT reflecting your Mothering Skills... believe me.
My Husband... has NO tolerance at all.... for putting any child to nap or to bed. He KNOWS... it can be very hard. He admitted it. That HE would not be able to do... that, like me.
all the best,
Susan