AC heater, bridge and MOSFET, possible?
One possibly unacceptable effect is that the control circuit is directly connected to the mains this way, and that touching any part of it may be lethal. Proper isolation helps, but it's cheaper to restrict the mains voltage to a part of the circuit. 

For a heater you don't need the 1 kHz PWM; its inertia is high enough that you can do it at much lower frequency. I would suggest to use an opto-triac instead of the FET. The triac will have a larger voltage drop than a FET, but for the FET you would have two diode drops too, so that won't make much difference. Also the switching of the FET will cause switching losses as the FET goes through its active region 2000 times per second.  



A MOC3041 has a zero-crossing detection circuit, which will give you each time half cycles on or off. This reduces peak currents in the heater and EMI. Instead of varying the pulses' duty cycle you can now control the heater by varying the ratio between half cycles on and half cycles off.

edit re target precision mentioned in comment
Milli-degrees precision is impossible, but not because the on-off control would be too coarse. Suppose you have a vessel with 1 kg of water. (You mention heating around a tube, which may suggest a streaming fluid, but then millīdegrees are even impossibler.) Heat capacity of 1 kg of water is 4.2 kJ/K, so 4.2 J will "heat" it by 0.001 °C. Switching on the 350 W heater for half a 60 Hz cycle gives 2.9 J, good for 0.0007 °C. If you would control each of the 5 heaters individually you can even add per 0.00014 °C, or 14 % of the required precision.  

So the coarse on-off control isn't the problem. It's distributing the heat evenly in the fluid, not to mention measuring the temperature. How can you get a to 1 milli-degree even temperature in a vessel (let alone a tube with heating wrapped around it)? Even a magnetic stirrer can't guarantee that. (Note that a magnetic stirrer will also raise the temperature!) And at 60 °C 1 milli-degree is 3 ppm, and to control it to that precision you'll have to measure it to better than 1 ppm. How are you going to do that?  

Phase control may give you a theoretically higher resolution, but there's no way to measure its effect, and without the required feedback you can't get the control precision either.