What type of motor should be used to handle a static torque (blocked shaft)?
Torque and current have a direct relationship in DC motors (brushed or brushless if driven appropriately). The torque provided by the brushed DC is not really smooth because the rotor and stator magnetic fields are only perpendicular once per transition, but the brush assembly is made such that there are no gaps in the torque available throughout the rotation. Regarding overheating, you'll need to work out the current required to compensate for the load torque, calculate the joule heating and do a simple thermal analysis with the rotor to ambient thermal resistance and the Max rotor temperature that can be both found (hopefully) in the vendor's data sheet. So it's not necessarily a no go. Especially given that you can include a heatsink to the motor in your analysis.

You'll have the same thermal analysis to run on the DC brushless. Don't think that a 10W motor can be run for long at 10W regardless of the torque produced and its speed, the efficiency is variable depending on the speed of the motor and usually the rated power is stated at rated speed, just like DC brushed motors. DC brushless are more expensive though because they have a somewhat complex electronic commutation circuit.

You could also go down the stepper route: if the load torque is lower than the holding torque then it is also an option, even more so if the detent torque (when OFF) is enough. Good thing is, you don't need a feedback sensor in a closed loop with this option (unlike all other motor types), only a home microswitch to know where to count steps from. You'll need a stepper controller though, but it is simpler and cheaper than the DC brushless one.

In both cases a gearbox can be used to increase the motor torque output to meet your needs (or from a different point of view, to reduce the motor torque required), just don't forget to include the gearbox resistive torque in your torque budget (i.e. to ensure the motor torque margin is high enough to absorb mother nature's randomness).

In my opinion you should strike the 3 phase AC synchronous machine (which is basically a DC brushless raw without controller) out, generating 2 or 3 sine of variable frequency and PWM amplify them is just not worth the hassle here, there are simpler alternatives. Personally I'd investigate first a stepper with a gearbox, but that's the quickest, not the cheapest option (DC brushed).