How to determine the right trip point of thermal cutoff for 1/4 hp 3A 115V single phase ac motor
It appears to me that the current rating for use with an inductive load is 3A. You described the motor as "3A (no load)." I would recommend that you use the sensor in series with a relay coil and use relay contacts to switch the motor. The relay contacts should be rated for 1/4 Hp 120 V AC motor duty.

The sensor needs to be embedded in the motor windings in the same w, in the same locations that the thermal fuse was. Even then, the temperature of the sensor may be slightly lower than the hottest area of the motor insulation. I suspect the motor insulation is rated higher than class A. I believe that class A is rarely if ever used in today's motors.

Assuming class A insulation and selecting a 120C trip point is probably fine for an inexpensive imported motor. Insulation life is a matter of time and temperature. Running a motor with class A insulation continuously at 120C would shorten the life to about 1/4 of the expected life. If that happens for a few minutes once in a while, the life of the insulation would probably not be drastic.