So the switch-mode constant current drivers shouldn't be pwm'ed?
So are you saying the ZTX851 would be better to use than a logic level mosfet but still not better than getting a constant current source with a dedicated pwm pin?
Can I build this circuit instead of buying a different pwm constand current driver?
ZTX851 isn't a good part. The drive requirement for 300ma Ic is quite substantial for a mcu, and if you cannot deliver that much current, you see Vce going up and power dissipation up.
According to his formula to find the resistor value for R3 I should be using a 1.5ohm resistor, do these exist?
Yes. Don't forget that the resistor will be a little heater, too. At 350mA it will produce over 1/2 a Watt of heat. I'd get at least a 2W resistor to give it a chance of not burning your fingers.