In this passage, "tolerably" is an adverb. According to the definition I learned as a kid, an adverb is a word that modifies a verb, adjective, or another adverb. In this case, "tolerably", an adverb, is modifying "accurately", also an adverb. In your suggestion, "tolerably", an adverb, is modifying "accurate", an adjective.
Availability means confident expectation—gained from “tolerably accurately” (p. 202) represented experience over a significant time period—that the attachment figure will be available.
Availability means confident expectation—gained from represented experience which is “tolerably accurate” (p. 202) over a significant time period—that the attachment figure will be available.
I personally think that the second one (your suggestion) sounds better. However, note that the author is quoting another passage in this point. It may be that the original author used "tolerably accurately" in their writing because it made more sense. Since this author is quoting them, it may be that this author wants to keep the original wording, so they structured the sentence you quoted in an odd way to preserve the original wording.