The sentence in question: "I met with a psychologist to see if I was on the autism spectrum but they told me that I was not."
"They told me that I was not." would be an elliptical sentence. Normally a subject complement would be needed. But this can be understood from context. In the given context the complement is "on the autism spectrum".
A comma may help this sentence. It is fairly long and it is formed of two part, both of which contain a verb. It's probably not absolutely necessary, but it would help the reader break up the sentence.
The word that is correct, but may be omitted with no change in meaning.
According to standard grammar, a comma is required.
The word but is one of seven words that are referred to as coordinating conjunctions. We have a great guide to coordinating conjunctions, but the short version is that a coordinating conjunction is used to link two identical grammatical elements (i.e. nouns and nouns, verbs and verbs, etc). One key rule of grammar is that coordinating conjunctions that link independent clauses must have a comma before them. If a coordinating conjunction is linking anything else, it doesn’t need a comma.
"I met with a psychologist to see if I was on the autism spectrum" and "they told me that I was not" would both be independent sentences on their own, so they are independent clauses. The word "not" is standing in here for the phrase "not on the autism spectrum".
