Hiya Alex!!
How's Matthew doing?? Funny that our LO's are the same age and I'm facing this same problem!! DS1 though is in school from 8:30 - 3:30. As such, I've worked him out on a wonky 4 hour EASY. He can stay awake for 2 hours, but when we're driving home from school, he gets dozy and then can't make it to 2 hours. I've found that after a week or so of fussing, he's getting better at transferring from carseat to crib if he's dozed for 10 minutes or so. And I sing LOUDLY in the car to keep him up as long as I can.
Here's our day ...
6:45ish wake - bf at 7:15
7:40 leave for school
8:30 falling asleep and home by 8:45 so I transfer.
10:30 - wake and bf
12:30 - sleep
2:30 - wake and bf - leave to pick up DS1
4:30 - catnap (sometimes this becomes 5 instead if he dozed on the way home from picking up DS1)
5:45 - wake and bf
7:45 - bath, top up and bed
10:30ish - df
3 am - feed
So not exactly what you're working with, but hope it kinda helps??? Maybe try some of those toys that you put on the back seat to entertain him? Loud music, talking??? Just some thoughts
Nish