I understand your reluctance in to increase the first A time however it is not really an increase, or not as much as you might think.
No matter when he goes to sleep he's up around 5-6 am! How do I get morning wake up later around 6/630 and get a better afternoon nap?!
You asked how to make him sleep later in the morning as the 5am WU is too early. Moving the first nap to 10am is a way of shifting morning WU time and making it more likely he takes a long nap every day rather than sometimes taking a shorter nap which throws off nap 2.
My suggestion is to shift things away from the school run. He might seem groggy and ready to sleep but in part that's because he's been awake since 5am and possibly also looking dopey from the car ride.
Here's how I think your day could look on the suggested shift:
WU 6.30/7
A 3.5hr/3hr
S 10.00 - 11.30
A 3hr (or might need 3hr 15)
S 2.30 - 4.00 (2.45 - 4.15)
A 3hr 15
BT 7.30pm
11hr night (which is roughly what you are getting now)
Although you've done a slightly later BT it's been after longer naps rather than later naps so I do think it's worth a full shift.
The difference with this also is that it reduces the A time before BT. You said even after a decent nap 2 he is awake by 3pm and BT is then later, at 7pm so there is sometimes a 4hr A time there, if this is too long for him (like is) then it can lead to EW the following morning.
If BT is earlier because he only had a 30 min nap 2.15-2.45 and BT is at 6.30 he is getting 3hr 45 A time before BT - again it can be too long
Whilst you are concerned about increasing the A time through my suggestions (but it does not go beyond the guidance A times if you review the sample EASY above) he is already getting longer than this due to EW and short naps.
He self settles especially for that first nap in the morning so there's no problem there so much it's that second nap that's tough and I don't know why.
Often it is not about moving just one nap but looking at the bigger picture of the entire day and night together. In this case I have looked over your times and the information you have given and tried to work out the best overall routine to aim for.
It is of course up to you what you choose to do