In this sentence
During this paksha, Moon is ahead of sun by an amount that is between 0 and 180.
What is the grammatical role of the word "that" in the sentence?
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M-W says that as a pronoun, that is
used as a function word to introduce a restrictive relative clause and to serve as a substitute within that clause for the substantive modified by the clause
- the house that Jack built
So in your sentence,
an amount that is between 0 and 180
that is a pronoun introducing the relative that is between 0 and 180, which modifies the noun amount. Within the relative, that replaces amount, being the subject of is.
During this paksha, Moon is ahead of sun by an amount [that is between 0 and 180].
This is a relative construction. The bracketed expression, "that is between 0 and 180", is a relative clause in which "that" is a subordinator' whose function is that of 'marker'.
Note: despite what most dictionaries say, and what you may read on grammar websites (most of which are not authoritative sources) "that" is not a relative pronoun.
One dictionary gets it right: link