Languages don't usually have names so on the nose as “Commonthroat”. Sure some official dialects have such names, like Putonghua for Mandarin, but those are introduced later.
Commonthroat was originally spoken around the River on Yih, so rDBGHqBdp “River speech” or rJhqBdp “River throat/River language” sound like decent autonyms, keeping in mind that, unqualified, the term qBdp “the River” refers to the equatorial river where the yinrih first emerged. There's also rDBGHsfqp “Yih speech”.
“Commonthroat” itself would be qBdqjHcp where qjHc means “pertaining to the whole” or “universal”.