An Adafruit PCA9685 breakout board (clones on ebay) and 16 logic level mosfets should be able to do what you want. The 220ohm series resistor is not a problem.
A small issue could be PWM whining noise from the motors, since the upper PWM range of the PCA9685 is 1kHz.
You could try other frequencies than the default 200Hz.
The PCA uses FastMode+ I2C. Several meters of cat-6 should not be a problem.
I use 6-7meters between a Mega and two PCA boards, running problem-free 24/7.
Leo..