Get an AAA or AA battery and briefly touch the LED ring leads onto the battery, red one end, white on the other.
Whichever way round the LED lights, the lead on the top of the battery is the positive, the lead on the flat bottom of the battery the negative.
With a 1.5V AA/AAA battery, you will not do any harm connecting it the wrong way round for a few seconds.
Edit: oh, hang on, just noticed the bit about the black in-line regulators. You won't get 1.5V through those, so will need to desolder or piggyback directly onto the LED rings from your AA/AAA battery with some spare wire.
The in-line regulators are very odd, not seen that before. Unless you have test gear, and not knowing anything about the white brick's quality or provenance, you should probably keep the black regulators in circuit in the final build.
I notice that the white brick's 12V output has a black lead on the +ve and a red lead on the -ve, so you are wise to be cautious.
Edited By John Doe 2 on 04/03/2022 11:27:40