Linear Regulator: Output current for stabilization different than datasheet?
I would be very suspicious of the 4.7uF capacitor actual value under bias. Try measuring it with a bridge that allows bias to be applied, or cob something together with a large series capacitor and a large value resistor to apply bias voltage (and check the manufacturer's data, of course). It's not uncommon for very small ceramic caps to lose 40% of their nominal value or more under bias. If you're in the unstable or conditionally stable region of operation, the loading will have an effect. 

Note Figure 13 from the datasheet:



As you can see, the minimum required ESR increases from close to zero at higher capacitances up to maybe 20m\$\Omega\$ at 2uF. You need to guarantee that the regulator sees a combination of capacitance and ESR inside the stable zone under all conditions (worst case is probably high or very low temperature and taking tolerance and aging into account). A quick test would be to slap a 50m\$\Omega\$ resistor in series with the capacitor.  

Using a physically large and higher voltage rated part may help (perhaps an 0805 25V part). Since you're using a CM you may wish to supply this particular cap if it turns out to be the problem.  

Edit: Since you've supplied the capacitor part number, I looked up the characteristics. The typical capacitance of this part is 1.6uF at 3.3V DC bias! 



That's at 25°C- if you need to operate over a wide temperature range you can't even count on that. And on top of that you have tolerance.