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Looking at your layout I have a few suggestions. If possible can you do the following:
From the waveform you've attached, it looks like the LMZ14203 is slightly in DCM mode. Using Equation 11 from the LMZ14203 datasheet, the current that borders the DCM/CCM boundary for your application is approximately 568mA. In DCM mode, you would expect higher output voltage ripple because of the lowered switching frequency as a result of minimizing switching lose. However, adding a 1k resistor would only result in and additional 1mA to your load which shouldn't have greatly affected your readings.
Can you provide me with the switching frequency waveform for both before and after the 1k resistor is added? Theoretically adding the resistor shouldn't push the output current more than 5mA which is still below the calculated boundary DCM/CCM current.
Also can you add another output capacitor with the same value to see if that helps regulate the output voltage? In parallel, we will get an EVM and populate it with your BOM to check the output voltage.
First I want to start by saying that in DCM mode, it is normal to see a bump on the output voltage. This is because in DCM mode the system is trying to maximize conversion efficiency lose as the frequency begins to drop which results in high output voltage ripple.
Secondly looking at the your first waveform without the 1kOhm, it looks like the load may be pulsing because there are two light load values every second during DCM. Every 4 seconds the load is increased and the system exits DCM and enters CCM and the output voltage improves. This can be seen when the voltage drops from 5.23Vdc to 5.08Vdc.
I wanted to see if this theory was correct so I took a LMZ14203TZ EVM and modified it to match exactly your BOM. I had an electronic load that stepped the load current between 50mA to 150mA at 1Hz, 50% duty cycle. Then I had an external 5ohm resistor in parallel with the electronic load that I manually inserted in and out every 4 seconds to simulate a sudden 1A loading to ensure the device exits DCM mode. It looks like I can duplicate the waveform from your document with the light load pulse and heavy load method.
Below is my waveform:
I am still unclear how a 1kOhm resistor could fix it since like I've posted before, 5mA added onto 425mA will not push the device out of DCM mode.