I think baby is over-tired. Keeping him awake in this newborn stage is actually having the opposite effect to the outcome you want.
Good sleep promotes better sleep. An over-tired baby will be really, really hard work to get to sleep and then will sleep fretfully so will wake up more easily. Whereas a just-tired baby will be much easier to get to sleep and will sleep more deeply and soundly.
In the first 3 months I wouldn't expect awake time between one daytime nap and the next to be more than an hour, probably more like 40-60 minutes at 9 weeks old. This might just be wake, feed, nappy check, quick cuddle and then straight back to sleep.
If you have a baby awake from 4pm until 2am with no sleep at all - that is a massive 10 hours awake. Baby must be so past being over-tired by that point. Then is clearly zonking from sheer exhaustion and it's messing all the rest of his sleep up.
What are you doing to get him to sleep? I favour dummy and bouncy chair when at home. Maybe swaddle, rock and into moses basket will work. Or pushchair/carseat for a sleep when on the school run.
If you have got to this extreme point of over-tiredness, initially you just need to focus on 'sleep, anywhere, anyhow'. Don't worry how or where it happens, baby just needs to get out of this over-tired cycle and start sleeping more regularly - however you can possibly make that happen.