Unfortunately, the 3880 won't execute a basic cleaning while any ink cartridge is showing a low ink warning.
I swapped in two new cartridges, did the cleaning, and everything is fine.
My mistake is that I forgot(or never knew) that I could swap the original low cartridges back in, in order to not waste the remaining ink contained therein.
Having forgotten about swapping back in, I sawed open the "low ink" cartridges to see how much ink remained.
There was about 15ml in one, and about 20ml in the other cartridge. Ouch.