Etymology of "blackguard rating" in the context of the British Army during the Crimean War

Solution 1:

Blackguard rating is not an official term; rather, Nightingale is describing her encounter with Barry as the most scurrilous scolding she had received in her life.

Etymonline has blackguard, a somewhat dated term, originating

1530s, scullion, kitchen knave. Perhaps once an actual military or guard unit; more likely originally a mock-military reference to scullions and kitchen-knaves of noble households, of black-liveried personal guards, and of shoeblacks. By 1736, sense had emerged of "one of the criminal class." Hence the adjectival use (1784), "of low or worthless character."

Rating here is used in a largely obsolete sense, meaning a reprimand; it should not be confused with the more-common modern sense of measurement or evaluation. To rate was

"to scold," late 14c., probably from Old French reter "to impute blame,

Solution 2:

A rating is a “sharp scolding or rebuke.” A blackguard /ˈblæɡɚd/ is a scoundrel, especially one who uses foul language. (There's even a verb form of blackguard meaning “to ridicule or denounce with abusive language.”) A blackguard rating is therefore an extremely abusive and probably foul-mouthed scolding.

Nightingale is saying that she has suffered many scoldings in her life, “more than any woman,” but none of them were as base and abusive as the one James Barry gave her on that occasion.