Tolkien gave various roots meaning “climb” in the 1940s, 50s and 60s. The first of these was ᴹ√RAP “climb (as a tree by clinging[?])” on a rejected page of verbal roots in the Quenya Verbal System of the 1940s, with a single derived (verb?) form ᴹQ. rampa (PE22/127). In 1957 Notes on Names (NN) Tolkien gave √RET “climb” as part of an explanation of the name S. Orodreth, translated there as “Mountaineer (= ✱mountain climber)”, the only place Tolkien ever explained this name (PE17/182).
The root √RATH “climb” appeared in a 1968-69 explanation of the name S. Amroth as “upclimber, high climber” along with other words like Q. rasillo “squirrel” and Q. rantala “ladder” (UT/245; NM/363, 367). In these same notes he said S. rath “street” was influenced by this root, as it “applied to all the longer roadways and streets of Minas Tirith, nearly all of which were on an incline” (UT/255 note #16); he also said Sindarin had no other clear derivatives of the root (NM/363). Tolkien briefly mentioned √RAP “climb” in these same notes as a root that Sindarin did have, but this section was rejected (NM/367). Finally √RATH appeared in 1970 green-ink revisions to the Outline of Phonology, but there the root was unglossed (PE19/89 and note #101).
Neo-Eldarin: For purposes of Neo-Eldarin, I think √RAP is best abandoned, but I think it is worth retaining √RET as well as √RATH (verb vs. noun?), especially since √RATH seems to be unused in Sindarin, with the exception of its influence on S. rath.