Why bunkering pump motor trip in the morning?

If you can successfully start the motor at other times during the day then the problem will be with what is different between the morning start and those other successful starts. It sounds like your co-workers may be correct in assuming that the oil viscosity is the problem since oil that has been sitting all night is thicker and cooler and will take more to get it moving. Thus, the motor is being started under load. Increased your starting voltage from 0 to 65% is a good start. If you are starting the motor under load, which it sounds like your are, then 65% is not unusual. Also, the suggestion of bumping the start time 2 to 3 seconds is something you should try. Most large unloaded motors will get up to speed in about 10 seconds. However, if the motor is loaded during startup, most soft starters have a "kick start" setting which is used when starting a motor under a load. Check the manual, but using the kick start during the first 1 to 2 seconds of your starting ramp at 75 to 85% voltage then dropping it to 50 to 60% along with bumping the total start time another 2 or 3 seconds (10 seconds minimum, 15 seconds maximum) will probably solve your problem.

Side note: Soft starters will trip out if the motor speed has not reached at least 85% of nominal by the end of the start ramp cycle. So, that is why your soft starter is probably tripping. The soft starter should be giving you a "fault code", use the manual to verify why it is tripping.