During the homework scene at the beginning when Miss Oddbod comes in, Miss Battleaxe actually turns to face Henry twice.
The place where the film is set seems to be large enough to have two private secondary schools but only one primary school so surely there are other options for schooling rather than the 3 shown. This is supported by the fact that the TV series shows another primary school in Henry's area.
The talent contest which Henry is excluded from doesn't actually seem to be connected to the school and so had no say in if Henry entered or not.