Trouble with DC/DC (switching) converter
Your load current and decoupling is inadequate to maintain the stability of the internal switching regulator feedback, and is causing unreliable startup.  

Things to try


Add more load
Switch input cap to 4.7uF
Add 1nF cap between V- and Input GND
Increase output caps to 1uF
Alternatively remove output caps.
Add 6uH input choke
See Below




You must ensure  at least 3mA of current on each rail, according to the datasheet, I would go a little more above this to 10mA load and go down from there. The datasheet says that the minimal load should be no less than 10% of the maximum power



The datasheet recommends a slightly different input and output capacitor arrangement plus a 6uH choke. With additional capacitance of 1nF between V- and input Ground. And a recommended 4.7uF of input capacitance. 





This device does not require any output filters, it already has them, but the datasheet is unclear, The way I read Table 2 and Table 3 is that it is recommended to have 1uF of output capacitance, but no more than 100uF per channel. 



I would remove them first. Although 0.1uf Should not cause any problems, then add them back in as 1uF parts.  

Additionally, ensure that your op-amp decoupling does not overload the device capacitatively, excess output capacitance will cause startup issues

All or some of these may be contributing to your startup issues



These DC/DC converter bricks are Integrated Switching Mode Power Supply (SMPS) Converters. That is how they achieve such small size, high efficiency, and stable output.

SMPS Converters generally use current and voltage feedback to establish the duty cycle of the switching device (PWM). 

These feedback loops are sensitive to load impedance changes and require you to maintain a minimum operating current.

When you roll your own SMPS, whether from discrete components or using an SMPS PWM Controller (Bring your own inductor and feedback loop) you tune this feedback for the size of the switching inductor, expected load, minimum load, and desired regulation/rippler

On other DC/DC bricks they may provide a feedback compensation pin to strap on more impedance to tune this behavior slightly, these all-in-ones do not, but will still be susceptible to the same kind of issues common to all SMPS designs.