How do I contain a 16-19A power on spike from a brush motor on a 10A converter?

An inductor perhaps? I'm not sure of how much inductance it'll require, but it has to be big enough to not saturate until 20A. Otherwise maybe an NTC resistor.

 

I'm not an EE, so my thoughts may be looney -- but I love problem solving...

You don't describe the converter, but if it's e.g. turned on by a logic signal, how about continuing to supply that logic signal until it's time the motor has spun up?

The choke is the first thing that occurred to me too, so that may be the most promising line of attack.

Would it be possible to have a capacitor charged and ready to help supply some of this transient current? 20 A for 2 ms is 40 mC; at 12 V, this requires a cap of (40 mC)/(12 V) or 3300 uF; that's not horribly large.

Is there a way to start the motor so it's not under load? You said accumulator, so it sounds like this might be part of a compressor or hydraulic system. Could a pressure relief valve be opened momentarily before starting the motor?

How about a low-resistance MOSFET in series with the load with a shunt resistor? An op amp would monitor the current and clamp it by increasing the FET's channel resistance.

 

Inductor currents rise linearly from zero with a rate of: I[A/s] = U[V] / L[H]
So for example: 12[V] / 2000[A/s] = 6[mH] which reaches 10A after 5 milliseconds at 12V applied.
But an equally important value is the current at which the coil saturates.
At that current/instant the inductance value becomes practically zero (the air-cored value). This current value is rarely specified.

The motor spiking at 19A means it has an internal resistance of 12V / 19A = 0.63 ohms. You'll want half the current and hence double the total resistance.
An NTC of 0.6 ohm initial (cold) resistance, and with a size which has a nominal operating current range encompassing 2A (or whatever) should do the trick.
Here and here are papers/ application notes from one manufacturer. An EPCOS S237 might fit the bill.

Daddles suggestion about using a MOSFET is also good. Just connect a resistor from +12V to Gate, and a capacitor from Gate to ground to control the startup time.
An op-amp controlled Gate drive to obtain an active actual current limited operation is also a possibility but might be more complex than neccessary.

 

Just as humble a suggestion as i can make - I'm not very familiar with this kind of problem...
My first thought was for an oversize bifilar inductor in the power leads and a high-current back-emf diode across the motor.

 

Too much resistance will make the compressor hard to start so select the lowest resistance, and the bigger it is (physically) the more power loss it'll have.
Try to look at All Products » Passive Components » Thermistors - NTC and then select (filter) 1 ohm for example. There should be many in stock that'll work for you.