Who was the high priest under David and Solomon whom the Sadducees claimed to be the spiritual heir of?

Answer 1

Abiathar

Mark 2:26How, in the account about A·biʹa·thar the chief priest, he entered into the house of God and ate the loaves of presentation, which it is not lawful for anybody to eat except the priests, and he also gave some to the men who were with him?"

Answer 2

The answer you're looking for is Zadok. However, the Sadducees were actually begun by a different Zadok; one who lived about 600 years later, in early Second Temple times. He was a righteous man, but his teachings were misinterpreted as if to support epicureanism (materialism; hedonism).

The Sadducees were men of politics and secular life, continuing in the ways of the Hellenising Jews. They had abandoned various parts of Judaism; and they claimed no earlier source or tradition for their attitudes. They harassed the Torah-sages; and, like the tiny breakaway group called the Essenes, disappeared at the time of the Second Destruction, just as the earlier Jewish idolaters had disappeared at the time of the First Destruction.

Note that there is a common conception that the Sadducees, like the much later (and now largely defunct) Karaites, made a deliberate decision to reject the Oral Law and reinterpret the Scriptures.
However, a careful perusal of the Talmud reveals that the Sadducees were actually opportunists who had nothing much at all to do with religion in any fashion. They were lax in Judaism; they were men of politics who weren't interested in Torah-matters.

At that time the Jewish courts still had the ability to enforce the Torah laws; and almost all Jews were Torah-observant; so, in order to avoid total rejection by the surrounding community, the Sadducees outwardly maintained a facade of keeping the major Torah precepts (such as the Sabbath), while ignoring the Oral Torah and customs and flouting the words of the Sages.
They went lost not long after.
The group that did (on rare occasions) debate against the Torah-Sages concerning subjects of religious observance, were a tiny sect called the Baitusim (Boethusians), who quickly died out.