Yes, Charlie B makes a good point...choice of wording is all the difference in the world when intrepting a system. The code defines a 3 hour load as continuous so the reason that's worth stating is just to make sure you are considering equipment based on continuous load when applicable but motors are a different world.
With that I'm not sure I'd agree with the 350kcmil without saying that you need to look at the specs of your motor in more detail because I don't necessarily agree that this solves your problem or if you even have a problem, though likely you do.
Obviously sizing larger yields less losses and a more efficient use of power but the only thing I can think of that gives us guidance in the NEC would be Article 695 referring to fire pumps - I would use that if the motor didn't give me a clearer indication of minimum startup voltage. Considering I would assume this is on the "safe" side of motor design I would think if you comply with that you are well within a factor of safety unless your motor says otherwise. I believe Art. 695 calls for 5% VD or less on 115% of full load amps (FLA) and 15% on starting. That's actually pretty hard to achieve at your distance so hopefully you have some better specs available.
If you consider a 10x inrush (many motors are more so consider that when considering my calcs below, some motors might be less too) with 2x250kcmil of copper (material wasn't specified before but that what I assumed you are planning to use) you are at 42% v drop and 4.8% on FLA. So yes I'd consider using larger conductor but that depends on what your motor needs. Some motors can run at 50% voltage on startup. Keep in mind this depends on what load is connected to your motor during started up, this works similar to a soft start in that you are reducing your startup torque, in this case to about 32% of full voltage startup torque (may be labeled as pull out torque or max torque) which if, for example, your motor is designed for 150% would leave you with 49% of normal operating torque - if you don't have any load connected until after the motor is in a steady state, maybe that is OK, maybe it isn't...again check the specs. Changing that the 15% VD would give you 72% of max torque and 108% of normal.
If you are sticking to no larger than 500kcmil CU then you'd need 3 runs of 500 CU to get you close to that 15% (it would give you 15.7%).
One question, what is the service voltage? I see you are from the USA and not Europe so is it 480V? If you are at 480v I'd presume you are using a transformer to change from 480v to 400v? If so, you should transform it by the motor if you have the space and means to do so. That being the case you would have a 16% drop using only 2 runs instead of 3...saves you an entire run for a small increase in VD.
General rule of thumb I am using with Vdrop is at least double the required ampacity of conductor for every 400ft (total conductor feet of the entire system end to end including intermediate devices, sub panels to the main panel, ect.). In my typical line of work I have to keep things less than 2% total system.
