What early-Indic word is the root of the English word "Narthex": a
west-facing vestibule characteristic to Byzantine churches, which is
derived from the earlier Greek word "Narthex" meaning "giant fennel"?

I expect that it is hopeless to seek an earlier root than the Greek
word itself.  Here is waht the etymonline website says:
"narthex Look up narthex at Dictionary.com
    "porch at the end of early churches" (used by penitents not
admitted to the body of the church), 1673, from Late Gk. narthex, from
Gk. narthex "giant fennel," of unknown origin. The architectural
feature so called from fancied resemblance of porch to a hollow stem.
The word also was used in Gk. to mean "a small case for unguents,
etc."