For me looks like the module is switching at about 70kHz and making the SMPS sag a bit at that frequency, can you see how the dips grow in the SMPS as you increase the load? Also, something similar happens to the module, where with no load there are no spikes, then at 10% load the spikes are there but the ripple ignoring the spikes is very low, and once the load increases more there is some ripple to be clearly seen in the 50mV/div scale which was barely showing off the noise in the 10mV/div scale.
It might be interesting to see how the SMPS behaves on it self, loaded with a easier load than the SMPS.
To filter the spikes you need a capacitor that has low enough ESR and ESL to deal with them, it needs to provide 5A, now you have a ripple of 200mVpp which suggests 40 to 20 mΩ ESR+ESL on the output cap, the sharp spikes suggest main problem is ESL. You should try using MLCC SMD caps, as they are shorter and thus have lower ESL. Zooming in even further, to check for the spike itself could throw some numbers in the capacitance and inductance shown in there.
Then you have the ringing between each spike, that's over the 70kHz, it can almost fit two cycles so about 140kHz, but as it's much lower than the spikes I'd leave them as they are till I get the spikes down. As the spikes are so fast, I wouldn't expect much inductance to be needed to deal with them, a few turns into a core in one of the wires coming from the module to the output should help, then dealing with the ringing it could need something different..
For the low inductance caps, I think the best way would be a double sided PCB, one side untouched as ground plane, the other with a slot in the middle for the cooper, so the caps go mounted side to side. You get into the PCB with the wires from the module and came the other way with the wires to the output or just mount the output directly to the PCB, holes to screw the banana jacks directly (you might want to have ±outputs side by side and the earth to the negative side as is sometimes done). In the PCB I'd install quite a few 0805 caps, or something similar, rated for 50V but with not much capacity, under 1µ for sure. 10 caps of 220nF would make 2.2µF which is a nice number for the output cap of a lab PSU while getting them distributed in a 50x50mm PCB as described will make seriously low ESL. You could even mix the caps, interleaving some 1nF and 10nF to go crazy with the 2.2nF ones. You don't need as many as the small ones as you are not trying to increase the overall capacitance with them but reduce the HF impedance of the solution.
In any case, even as fun as this is and informative, you shouldn't expect this SMPS to be the low noise PSU from your lab, but the flexible one for the odd load. Having that output range (V and I) will make it's use inside the lab even being this noisy. I still believe some improvement could be made.
JS