If Kaneki had eaten another part, I believe the damage to Hide would be much greater, possibly leading to his death.
Maybe the mouth was the best option
"Hide lets Kaneki eat his face to get back his strength. While it appeared at first that Hide had died in the process, he reappeared later as Scarecrow, an ally of Kaneki and the ghoul hunters."
Already answered in comments by @W. Are
Hide never died. The anime deviated from the manga and the scene where Kaneki was carrying Hide never happened in the manga. My suggestion: if you have time, read the manga (starting from Tokyo Ghoul). I tried watching the anime, too but dropped it while on Tokyo Ghoul: re. It was confusing and the manga cleared up ...
Tokyo Ghoul √A isn't canon, meaning the story doesn't follow the manga. Tokyo Ghoul Season 1 ends somewhere in the middle of volume 7, so to be able to follow Tokyo Ghoul: re, you'd have to finish the remaining volumes(7-14).
I think there never was that kind of interaction, beside
(I would love to provide more to that scene, but im not that familiar if it is possible to tag pictures as spoilers so i hope that the episode is enough as source)