Say I do some bus rides in the morning and lot of tube journeys at off
peak time. Is Oyster intelligent enough to handle this?

I'm not sure that I am following your logic as there is not a separate
peak cap for one day bus travel. Regardless of when you travel you will
cap bus travel at £3 - you simply get to the cap a bit faster by using
buses at peak times at £1 per journey.

The increments on from £3 to the respective off peak (ODTC less 50p)
caps are as follows for off peak tube travel (i.e. after 09.30)

£1.40 Z12
£1.90 Z14
£2.80 Z16

As far as I know an off-peak paper day travelcard is not valid before
9:30. Not on the tube and not on the bus. So if one wishes to travel by
bus in the morning this has to be payed separately. Right?

So my logic was that it is the same with Oyster. If the Oyster off-peak
cap allows peak bus rides -as you say- this would be a reason more to
use it.

Martin