I wonder if that first short nap is throwing the rest of the day off? I appreciate it is on the school run but Is there anyway you can extend that I.e walk or drive for longer even to get 30-45 mins. That might mean he could manage a longer 2nd A time and longer nap middle of the day?
At this age I think much DS was doing something along lines of
A=2.5-3hrs (Milk and breakfast)
Nap: 9-9.30/45 (he woke himself after short nap)
A= Around 2.5hrs (Milk, play, early lunch)
Nap : 12/12.30ish for 1.5-2hrs
A=around 2.5hrs (Milk, play)
A= Around 2.5hrs (Tea, milk)