I hate to say that the whole premise of your story is a little off.
Ilyusha is the normal diminutive for Illya, and has no demeaning or belittling connotations. It is what close friends and family would normally use as address. By the same note, there is nothing girlish about "Napasha" or "Polya" as an affectionate or familiar term of address.
It was a bit forward of Mme Grushenka to address him as Illyushenka, which is the child's form of address--or that used by VERY close friends or family.
The log of wood remark is funny, but falls flat if you know the correct terms of address, which removes the humor.
IK has a very snarky sense of humor and would probably find something much more devastating if he were truly annoyed--which of course he would not be since he has probably been addressed either that way or by his given name and patronymic his entire life. It's the American version that is stiff and formal.
From: periwinkle Date: 10/23/2008
I loved it when you first thought of it, and I still think it's a hoot.