The lamp requires more current than the converter supplies
Unlikely, given the ratings shown.
The cap is too small to supply adequate smoothing/charge and/or is not the right component to solve this problem.
Could be. 220µF isn't a whole lot of capacitance when trying to filter DC power. Could try a much larger value, such as 2200 or even 10000µF (0.01mF), but this may lead to instability described below.
The black box "converter" has additional circuitry (which the others don't) that deals with the flicker issue and I should add this (what?) to my adaptor/connector.
There are several scenarios here.
If the original AC/DC converter was a bulky linear supply (bigger, mains-frequency transformer) then it likely output fairly smooth DC, resulting in no observable flicker.
If the original AC/DC converter was a small switching supply (smaller, high-frequency transformer), made specifically for this lamp, then any "ripple" was likely not visible, as it is in the kilohertz range (thousands of times per second.)
With a switching supply, adding too much capacitance to it's output will alter the behavior and can lead to instability, oscillation, even "hiccup" operation.
Additionally, switching supplies designed for a max current of 1.5A, when run with a much smaller load, can adapt to this reduced power demand in several ways. They can reduce the switching frequency, or "skip" pulses. Both have limits, and either of those can result in increased output ripple, which could manifest as visible flickering. Powering LEDs from switching supplies with zero measurable flicker is actually a challenging design goal.
If the original supply is rated for 15V, then the replacement should be also. An LED array designed for 15.00V may barely power on at all at 12V, drawing far less current, and exaggerating the intermittent switching behavior.
Either find a supply more closely-suited to the LEDs (such as 15V 0.4A) or consider getting a >=17V supply and adding an LM317 adjustable linear regulator after it.
The LM317 will have to be mounted to a heatsink as it will lose about two volts across it. But it will "regulate" the output without any visible flicker (even if the power supply had lots of ripple.) An example of using the LM317 as a constant-current regulator can be found here or search this site for "LM317" for tons more. Just choose an R1 value to set the desired brightness (current); 0Ω would be max, use 1/4W or larger. Note a potentiometer cannot be used in this configuration. And note this has the added benefit of relaxing the power supply specs somewhat... any-old laptop charger (assuming 19V or 21V) could be used for instance (careful not to burn out the LEDs with too much current!) But the higher the supply voltage goes, the hotter the LM317 will get, so there are limits.