Hmmm, it's hard to know what to do, I can feel how desperate you are to get started with sorting this out but if he's still not right......really it's not fair to expect him to learn too much yk? Is the LC helping you with feeding/positioning and latch? I wonder if you wanted to do something whether you could very gently start to try to break the sucking to sleep association using the gentle removal plan?
Gentle Removal Plan It doesn't feel like the right point yet to be doing anything too drastic, what do you think?
Routine wise although he's tired, you may need to get pushing the A times a little and try to just have three naps unless they are all a complete disaster. He may well be uncomfortable at night and waking because of that, but too many sleeps/too short A time really won't help the situation either. It will be hard to increase though because he will be tired from his broken nights, but probably you need to get to at least 1h45 or so if not 2h. It's good you can resettle him

. Overall I'm wondering if you could aim for him to fall asleep on the school run around 9ish for his first nap (will he transfer?) and then if it's a long one do the second one at 12.30/1ish, to be awake for the school pickup, then a catnap in the evening if he needs one? If it's a short one then down for second nap earlier and hope for a longer 2-2.5h one?
Something like:
WU 7, nap 9-10.30, nap 12.30-2/2.30, CN around 4.30, BT 6.30/7
Or WU 7, nap 9-10.15, nap 12-2/2.30?
What about trying to roughly establish a routine first then focus on the independent sleeping afterwards once you've seen the paediatrician?