What is the correct way to handle inrush currents from a DC/DC converter with short-circuit protection?
Either of your solutions sound entirely reasonable. There are some power supervisory chips that could handle the turn-on delay. For example, the inexpensive ADMxxxx series:-



\$ \overline{ENOUT}\$ on the ADM1087 is an open-drain output that could directly drive a P-channel MOSFET gate (with a pullup resistor) and source connected to +12 to give you a high-side switch. You only need to select the delay time (by a capacitor) the threshold (by a voltage divider pair of resistors) and give it a low voltage supply voltage such as 3.3V. 

If the DC-DC cannot handle the capacitive load that is required for reliable starting (most have limits specified in the data sheet, sometimes as low as 200uF), then this is a scheme that I've used before in much more demanding situations. It requires two switches (or to use an enable line on the load). The Big Fat Capacitor (BFC) charges through R1 and when the voltage across Q1 is sufficiently low (as determined by a supervisory circuit such as the ADMxxx measuring the voltage and time), both switches close, shorting R1 and supplying power to the load. R1 sees some high dissipation potentially, so a pulse-rated resistor is called for. 







simulate this circuit &ndash; Schematic created using CircuitLab

In some cases, you may be able to replace Q1 with a diode and/or Q2 with an enable line on the load circuitry, so only the supervisory circuit (to drive the enable line), a resistor and a capacitor are required.

In other cases where mains voltage and hefty capacitors are involved, wirewound resistors and mechanical relays or Solid State Relays (SSRs) can be used, but the concept is the same at the kW level. 

NTC surge eliminators can be used in such (mains power) situations, however they drop voltage, waste power and are not reliable at their task (the capacitors can discharge faster than the NTC cools, then you get a huge surge if the power blips momentarily). They'll also not work well if your circuit needs a surge to get going, they will cause harm by giving it a voltage that is ramped up rather than a sharp startup. 

Of course if you can find a DC-DC that can reliably handle the startup surge, that will be simpler, but guaranteeing that may not be that easy (testing over temperature perhaps with two displays attached) and you might want to protect the DC-DC with a polyfuse.