I'm going to guess the 3.3V power supply isn't "holding-up" under the load. Check that. You'll probably need a "bigger" power supply or a different voltage regulator.
If you're using the on-board 3.3V regulator, your motor is rated for almost 1/2 Amp and that's probably too much for it (depending on how much voltage you are dropping across the regulator). If the regulator is too hot to touch, that's a bad sign.
If I measure the voltage across the motor, when it is running, the voltage never exceeds 1.5, no matter how high I set the PWM pin
You can't reliably measure the PWM average with a digital multimeter.* But, you can (briefly) write a digital high (or 100% PWM) to see what you get. (An analog meter movement will give you a good average reading.)
If I use 3.3/255 = 0.013, the 1.5 volts should be reached when I set the PWM pin to 115,
Once you get it working properly you can boost that up a bit if you need to. You'll get a fraction of a volt drop across the transistor and although I didn't see a maximum voltage rating for the motor, there is a graph that goes up to 1.8V.
* P.S.
You can check your meter:
Disconnect the motor so there's (almost) no load.
Write a high or 100% PWM and read with the meter.
Write 50% PWM can read again.
...If it's stable and close to half of the DC reading you can use your meter to read PWM. You also can try something like 10% & 90% to see if the readings are linear.