If going belt driven you will first need to add a flexible coupling to the generator input shaft, then have a pair of bearings in pillow blocks mounted to drive it, and then the pulley for the belt or chain drive. Reason is because most alternator drive shaft input bearings are not rated for large radial loading, they merely are there to align the rotor with the stator, and the bearings do not have the load ability to handle your belt tension plus the torque the belt or chain will create on them.
You will need the coupling to isolate the radial torque, and the pillow blocks, with appropriately sized shaft, generally a few sizes bigger than the input shaft for stiffness, and the pillow blocks to handle this very large load. Your pillow blocks will handle the radial load perfectly fine with the larger bearings they have, and in general you sort out most of your alignment there, leaving the flexible coupling to take up the rest, plus any vibration from the belt or chain. All standard parts available off the shelf from mechanical parts suppliers, including the ready made stub shafts with keyways already cut.
Same for the car engine output, the bearings there are also not rated for radial load, and need the same treatment. the crankshaft axial load rating is there only for the thrust from the clutch or torque converter, and radial loading capacity is there only to keep the crankshaft centred, as the set of main bearings in parallel absorb the thrust of the pistons only, they have an oil film that will only handle that.