OP:
So my question is, why did John call them "brood of vipers"? Now, I understand the application of this title to these two groups overall, but why specifically in this instance when they came to be baptized?
The assumption is that they "came to be baptized", but did they? Is that why they were there at the Jordan?
John preached the baptism of repentance for the remission of sins (Matthew 3:6, Mark 1:4, Luke 3:3).
Is that why the Pharisees* and Sadducees* came to John? Was it to have their sins forgiven? If it were, it would have been a clear direct denunciation of their own temple service. It was in the temple were via animal sacrifices that their sins were removed (remitted).
Remission means "Deliver, Deliverance, Deliverer:
denotes "a release, from bondage, imprisonment, etc." (the corresponding verb is aphiemi, "to send away, let go"); in Luk 4:18 it is used of "liberation" from captivity (AV, "deliverance," RV, "release"). " -source-
Both sects were active in the temple within "The religious responsibilities of the Sadducees included the maintenance of the Temple in Jerusalem. Their high social status was reinforced by their priestly responsibilities, as mandated in the Torah. The priests were responsible for performing sacrifices at the Temple, the primary method of worship in ancient Israel. and without. " [T]he Pharisees continued a form of Judaism that extended beyond the Temple, applying Jewish law to mundane activities in order to sanctify the every-day world. This was a more participatory (or "democratic") form of Judaism, in which rituals were not monopolized by an inherited priesthood but rather could be performed by all adult Jews individually or collectively; whose leaders were not determined by birth but by scholarly achievement."
The point is John and thus Jesus represented to them something new and different. They were, to some in the sects, the outsiders trying to challenge their positions, prestige, and pocketbooks. They were not there to be baptized, but to do reconnaissance.
To be sure, there were others, like Paul, Nicodemus, Joseph of Arimethea, who did become baptized and did believe, but they left the "brood of vipers".
Pharisee "A sect that seems to have started after the Jewish exile. In addition to OT books the Pharisees recognised in oral tradition a standard of belief and life. They sought for distinction and praise by outward observance of external rites and by outward forms of piety, and such as ceremonial washings, fastings, prayers, and alms giving; and, comparatively negligent of genuine piety, they prided themselves on their fancied good works. They held strenuously to a belief in the existence of good and evil angels, and to the expectation of a Messiah; and they cherished the hope that the dead, after a preliminary experience either of reward or of penalty in Hades, would be recalled to life by him, and be requited each according to his individual deeds. In opposition to the usurped dominion of the Herods and the rule of the Romans, they stoutly upheld the theocracy and their country's cause, and possessed great influence with the common people. According to Josephus they numbered more than 6000. They were bitter enemies of Jesus and his cause; and were in turn severely rebuked by him for their avarice, ambition, hollow reliance on outward works, and affection of piety in order to gain popularity."
Sadducee "Sadducees = "the righteous"
a religious party at the time of Christ among the Jews, who denied that the oral law was a revelation of God to the Israelites, and who deemed the written law alone to be obligatory on the nation, as the divine authority. They denied the following doctrines:
resurrection of the body immortality of the soul existence of spirits and angels divine predestination, affirmed free will
