Surprisingly hard. Delousing was a major endeavor all the way up into the 20th century. The little bastards are hard to completely eradicate.
I did some searching, but turned up kind of empty. Do you have something you could link on this?
Sure thing! This is about WW1 in particular, but it gives an idea of how hard the little bastards are to fight.
https://spartacus-educational.com/FWWlice.htm
Where possible the army arranged for the men to have baths in huge vats of hot water while their clothes were being put through delousing machines. Unfortunately, this rarely worked. A fair proportion of the eggs remained in the clothes and within two or three hours of the clothes being put on again a man’s body heat had hatched them out.
So how did that machine work? My search turned up a lot of stuff about it being a problem in the great wars, but urban Rome is a very a different beast, with much less science and much more leisure time and stability.
My first instinct is boiling water. The Romans would definitely have tried that, and few things can survive it, but I’d also guess their textiles wouldn’t have survived well. My next thought is smoke, which I guess they could have just missed, but then again maybe the pests are resistant to it. The eggs at least wouldn’t need to actively respire much. Salt maybe? Tiny things don’t generally deal well with the wrong salt concentrations.
In Auschwitz they used the same cyanide gas tablets as were used on the “prisoners”, but in much greater concentrations, which has become a source of supposed gotchas for denialists.