Stop the formal activities or at least reduce them. There are not necessary and they are probably making him more tired even if he enjoys them whilst he's there. By all means get out of the house every day, but go to the park or something, something that is more flexible for all of you.
And as for the new baby not bothering him, well I think that's highly unlikely. His whole world has just been turned upside down ... Mummy went away (to hospital) for ?a few days and didn't look like his normal Mummy for at least a day or two (pale, tired, etc) and then there's this little baby in the house, crying all the time, getting lots of cuddles & attention from Mummy & Daddy, etc, etc, etc. I think that the new baby is very much going to be having an influence on his behaviour. It doesn't necessarily make it easier to manage, but perhaps easier to understand?
My ds was anxious and withdrawn in the months leading up to my delivery and we weren't sure if it was pregnancy related or not. It was and he was a lot better after I had bubs. But then at about 6 - 8 weeks after delivery he started acting out and deliberately misbehaving too. It settled with time and lots of extra (good) attention ... but mostly he just needed time to adjust to the 'new norm' of having a baby around (who he adored from first sight & still does).
I suggest encouraging rest times (eg. stories/etc) to help him feel more rested. I also suggest some extra one-on-one time for him & you when possible (my mum said my brother could never have enough of that and it drove her nuts ... no amount was enough for him ... every child is different in that respect). And wait.