Lamentations mentions an "adversary" 5 times through the book. Only the book of Psalm has more references to an/the adversary. All NASB95: Lam 1:5: "...Her little ones have gone away as captives before the adversary" v7:"...her people fell into the hands of the adversary and no one helped her" and in 4:12 it seems to differentiate between 'the adversary' and 'the enemy':"...the kings of the earth did not believe, nor did any of the inhabitants of the world, that the adversary and the enemy could enter the gates of Jerusalem". Is this speaking of the satan?
Although "adversary" is sometimes used to describe "the satan" (ha-satan), here it means simply "foe" or "enemy." Lamentations is traditionally ascribed to the prophet Jeremiah, and its context is the aftermath of the fall of Jerusalem to Babylon. According to the introduction in the NABRE:
The Book of Lamentations is a collection of five poems that serve as an anguished response to the destruction of Jerusalem in 587 B.C., after a long siege by the invading Babylonian army.
Although the name of the adversary is not mentioned Babylon's actions fit very well with several of the verses mentioned in the OP. Captives have been forcibly exiled. Her people [Jerusalem's] have fallen into the hand of the adversary. Most tellingly:
1:10 The adversary has stretched out his hand over all her precious things, For she has seen the nations enter her sanctuary
Taken individually, these verses could refer to other historical events, but collectively they could only refer to the fall of Jerusalem to the Babylonians, when the nation collapsed, its citizens were forced into exile and the Temple was despoiled.
The notable exception to the identity of the adversary is found in 2:4, where it is the Lord who is Judah's adversary.:
He has bent His bow like an enemy; His right hand is positioned like an adversary, And He has killed everything that was pleasant to the eye. In the tent of the daughter of Zion He has poured out His wrath like fire.
Conclusion: The adversary mentioned in NASB95's translation of Lamentations is Babylon. The exception is ch. 2:4, where it is the Lord. In none of these instances does "adversary" mean "ha-satan" per se.