Dear Steven,
For what it's worth, it perhaps may be useful for you to know that حرف is used in the Arabic version of Aristotle's Rhetoric (BnF ms ar. 2346, composed by Ibn al-Samḥ - d. 1027) as a translation of ܐܣܛܘܟܣܐ (ʼsṭwks) < Gr. στοιχεῖον (even though the input form for this loanword was likely στίχος or στοῖχος, but this is beyond the point). The Greek word means 'element', 'first principle', so in rhetoric it is usually understood as 'a certain prescription or principle of the art of rhetoric'. In the Arabic translation of Aristotle's text, حرف is almost always used as a synonym for the loanword اسطقسات (in the plural). This could be understood as a reader-oriented translation strategy, perhaps prompted by the fact that στοιχεῖον has among its meanings that of 'letter', exactly like حرف.
Given that David Taylor's and Salam Rassi's hypotheses are, in my opinion, absolutely right and impeccable, I wonder whether this convergence of meanings, in which حرف is used to translate something that could mean 'a simple part', provides some further support for the reason why حرف has been chosen over other words in the context. Additionally, the Syriac word petgāmā, to which David and Salam referred, is used in the Syriac rhetorical handbook written by Antony of Tagrit in the ninth century with the meaning of 'line', 'verse'. The reason why I'm bringing rhetoric up in this context is that Ibn al-Ṭayyib wrote a Long Commentary (tafsīr) to Aristotle's Rhetoric and a compendium, so he must have been familiar with such technical terminology.
I hope this helps!
With best wishes,
Mara