What is the current limit of driving a linear actuator for 4 VDC at 12 VDC?
This will depend on your specific motor driver a little, however if you can set a current limit on the driver, you can use the lower voltage device with a higher supply voltage, your driver will get a little hotter depending on the method it uses, but it also will give your actuator a higher maximum torque and higher top speed in general. 

This is quite common in larger stepper motors, the motor itself would cook if fed with a constant 48V, but by having the controller limit the current once the inductance is overcome it runs fine. the perk being a higher voltage gives a higher torque,

When I played with this in the past I was dumb enough to use a linear current limit off a 72V supply for 12V steppers, they absolutely flew, and could almost stop on a dime, and while the motor stayed fairly cool, the current limiting transistors where practically glowing after a few minutes

In your case, It would apply 12V, the inductance of the phase would take some time to be overcome, so during this time the voltage would be fairly constant with the current rising, as it approaches the current limit, the voltage will fall down to the resistive voltage drop for the given current, and you would end up with about 4V * 0.2A on that phase, I would assume its a switching controller, so you would be drawing ~12V * 0.07A on the input (essentially a buck converter) at this point, and may fold back to a lower holding current, again based on what your controller is set up to do.