A couple of other things to check...........
Is there a difference between Battery start and GPU start? This will eliminate the problem of a "weak" battery. On the warm start the battery will be at a higher state of charge as compared to cold.
One obscure thing to check is that the excitation of the GEN side of the Starter/Generator is isolated during the start i.e. no field voltage. The way this is done varies between airframes. The generator is always generating and is normally isolated from the battery and main bus by the RCR. During the start though the field excitation needs to be isolated so the SG is not trying to start and generate at the same time.
The other quick test of course would be to throw another FCU at it and see if there is a difference.