does a choleh have to eat less than the shiur on 9 Av?

Aruch haShulchan writes in 554:7 that someone who is ill on 9 Av does not have to eat little bits at a time, each less than the shiur, like on Y"K

פחות מכשיעור לא שייך בתשעה באב, שהוא מדרבנן.  

This is not the same case as the Biur Halacha (on 554:6) we discussed a few years ago who writes that when there is a danger of cholera, a person should eat less than the shiur.  In that case, the person is not yet ill -- there is just a great potential of them becoming ill in the future.  Therefore, B.H. advises to eat less than the shiur, which is not considered achila, so that the day still has a shem taanis.  The Ah"S is talking about where someone already is ill and therefore cannot fast.

It's not clear exactly what the Ah"S means when he says the difference between Y"K and 9 AV is that 9 Av is derabbanan.  You could read it as suggesting a chiddush that is often quoted in the name of R' Chaim Brisker, namely, that when the Chachamim instituted the taanis, they were never gozeir on those who are sick.  For someone ill, the day does not have a shem taanis, and so they gain nothing by fasting. 

(I saw R' Mutzafi goes so far as to ask whether someone who is ill and exempt from fasting has a chiyuv melaveh malka since the day is not a taanis for them.  He writes that the Chachamim did not institute a chiyuv m"m on 9 Av, but the point is that the question clearly assumes like R' Chaim.)

The Marcheshes (siman 14) disagrees.  He writes that he took ill one 9 Av and he ate in bits less than the shiur because 9 Av, as divrei kabbalah, is as stringent as Y"K.  

The difference between R' Chaim and the Marcheshes could boil down simply to whether we treat divrei kabbalah as more stringent than a regular derabbanan or not, an issue that comes up in other sugyos with respect to krias ha'megillah and halachos of Purim.  

Or it could be that even if you accept that 9 Av is treated as a regular derabbanan, you can still take issue with R' Chaim's chiddush based on the principle that kol d'tikun derabbanan k'ein d'oraysa tikun.  If the d'oraysa of Y"K requires eating less than the shiur when one is ill, the takanah derabannan of taanis may parallel that.