The verb ᴱQ. ter- appeared untranslated in a list of Qenya Verb Forms from the 1910s (PE14/28). Patrick Wynne and Christopher Gilson suggested it might be based on the early root ᴱ√teře “pierce” from this same period (PE14/28 note #4; PME/91).
Conceptual Development: The root √TER “pierce” appeared several times in Tolkien’s later writings (Ety/TER; PE17/33), so I would use ᴺQ. ter- “to pierce” for purposes of Neo-Quenya. For example, Helge Fauskanger had the active participle ᴺQ. térala “piercing” for his NQNT (NQNT). I would use the active participle form terila myself, but I think the basic verb ter- is perfectly viable.
A noun appearing as ᴱQ. teret (terekt-) “auger, borer, gimlet” in the Qenya Lexicon based on the early root ᴱ√TEŘE “pierce” (QL/91; PME/91).
Neo-Quenya: The root √TER “pierce” appeared in Tolkien’s later writings so I think this ᴺQ. teret “auger, borer, gimlet” can be salvaged for purposes of Neo-Quenya, but I would update its stem form to tereht- for consistency with later Quenya phonology.