How to choose proper motor for agitator?

What motor rating is used is determined by the amount of torque required to move the load. Horsepower does not move the load, torque moves the load. A 1-hp, 6-pole motor will develop 50% more torque than a 1-hp, 4-pole motor If the manufacturer of the agitator is provided with the density of the material beforehand they should determine what torque is required and supply a motor to suit.
If the original motor is going to be used it may be necessary to shave some material off of the agitator blade.

Check with the motor manufacturer if the motors are thermally suitable for a continuous (or X seconds) overload of 115% to 120% and then set the overcurrent protection relay to, say, 120 or even 125% FLC (preferably with the time delay of X seconds) to account for sudden overload common in agitators. This should avoid nuisance trips as long as the motor thermal capacity is sufficient since the breakdown torque must be at least 150%FLT.

In my opinion, the agitator manufacturer should have guidelines for establishing the driver size (and maybe gearbox too) for different materials. Especially recognizing different densities or other liquid properties (ie: lumps, clumps, solids, etc.)

Motors are designed to have optimum continuous performance somewhere between 80 and 100% load. Not all are the same. Some peak early, some peak at full load, some are nearly flat. But all of them decline in efficiency after full load. Higher output is possible, but at a higher watt/kW basis (and a higher winding temp).

Many severe duty users will preclude the use of a motor in the Service Factor range.
And at least for MV and HV motors, IEC does not allow for anything other than 1.0 SF.
I know of several MV users who will require F insulation, B temp rise @ 1.15 and still demand that the motor be sized to run at 90% of the nameplated Hp. Why? Because they know their operators will run up the load years later, no matter what the original design parameters were.
"We build 'em, and operations tears 'em down"

All this to say that a properly sized motor running at or below 1.0 SF should have lower amperage than one operating above rated load, the relay will need to be changed.

Rewinding to a higher insulation level is only a band-aid. If the original space and mounting could only accommodate the specific motor frame, then there would be no choice. Otherwise, I would go with installing the right motor the first time.
There are more considerations in motor design than just the stator temperature. Rotor cage and bearing temperatures are also impacted.