What is the industrial equivalent of a dimmer switch for resistive heaters?
1.5 kW ... 12 A ... I am trying to find a way to limit the current requirement of the heater so that the on state is ~5 A. 




Figure 1. Phase angle control. By delaying the turn-on point in the mains half-cycle power can be reduced from close to 100% to close to 0%.

You have correctly realised that even with phase-angle control set to a point to reduce the current you will need a device rated for the full current. For example, if the phase control of Figure 1 is set to 50% or 6 A in your application the peak current switched remains at the same value as a full-wave 12 A. There will be a small reduction at 5 A.

You have been caught out by one of the earliest pieces of electrical legislation - Ohm's law. Since \$ I = \frac{V}{R} \$ your only to options to reduce I are to (1) reduce V or (2) increase R.

Solutions:


Use a higher rated SSR. It won't complicate the controls any further but might may take a little more space. You need a non-zero-cross type but now need to time your trigger pulses relative to the mains zero-cross. This may complicate things too much for you.
Use a transformer to reduce the voltage to 5/12 of the original. A simple SSR will work then.
Change the heater to a 600 W, 120 V unit.
Use your full 12 A but with the required duty cycle to get the power you require.




Figure 3. Proportional on-off time control. Note that step size is one half-cycle minimum. This can make the response seem coarse if the repeat time is short.

Since most heater applications have some thermal inertia a slow (relative to mains frequency) on-off control can give the required average output power.