Hello, welcome back to BW forums
From what you've said I think it might be worth a shot at a more "set naps" kind of routine with a couple of CNs and a couple of longer naps rather than a standard 3 long nap routine. I think it might help reduce your worry about trying to get him to sleep longer all the time. What do you think?
The first A time is a bit too long for his age but clearly he has to wait until he's on the way to school to get that first sling nap so I doubt there is much you can do about that. I would probably just plan that as a CN, then plan a long nap at home, a CN before or during the afternoon school run and another long nap at home followed by A time then BT. If you set this up each day your LO could get into the habit of it as he'd know when he is going to get his sleeps, I'd suggest keeping to the routine on weekends too to help him establish the routine and know when he is getting the sling naps and when a cot nap.
So it might look something like this:
WU 7.00
E 7.30
A 1hr 45 a bit too long for age
S 8.50 - 9.30 (40 min CN)
E 10.30 (do not feed to sleep)
A 1hr 30
S 11.00 - 1.00 (2hr in the cot)
E 1.30
A 1hr 30
S 2.30 - 3.10 (cot or sling 40min CN)
E 4.00 (do not feed to sleep)
A 1hr 30 (including school run)
S 4.40 - 6.40 (2hr in cot)
E 6.40 before bath if needed
A 1hr 20/30
E 7.30 BT feed
BT 8pm
I think once you plan those CNs in you could feel less obliged to try to extend every nap, make use of the A time to get him fed or dinner made and the 2hr home nap in the evening should free you up to spend time with your older child, cook/eat, whatever. At the moment it looks like you are spending the entire evening trying to get LO asleep but with a more "set" routine you would only try to sleep train and resettle during the time he is to long nap. Initially it is not likely to give you additional Y time to spend with your older child but once the habit is establish I would imagine it should help.
By "set" times I don't mean you'd keep these times for ever, more like they are set for now then change as LOs needs change and then re-set again on a longer A time. You could for instance see that first sling CN starting later and being cut shorter by arrival home until it is the first nap to drop and instead the longer nap in the cot becomes the first nap of the day and may be brought earlier. Hope that makes sense.
In terms of sleep training I would suggest beginning with nap 2, stop the feeding to sleep and sleep train in the cot using shush/pat. As that becomes a little more successful then work on nap 4 to get that established in the cot. At the same time work on independent sleep for night sleep.
What do you think?