Dont think repair manual will help with your issue. Had a customer who took there car to toyota a few months back with a few issues, they recommended ECU so he called me up, I told him its not ECU but toyota insisted it was, he bought one and it didnt fix the problem, then they said it was loom, told him not likely but he bought one didnt fix the issue. Anyway after 4 weeks of Toyota having the car, I spoke to him on the phone for half an hour and the next day the car was fixed.
Really needs going over properly as something probably isnt right from when the engines been swapped.
The secondary air pump - if thats having issues it puts a large load on the battery - when they go bad they normally draw too much load and blow the fuse. Need to test the pump first and then check the valve from the manifold, and also check the vac pipes. But it might not be related to your issue. But given that the air injection only comes on when the car is cold it does kind of make sense...