The jokes about Arthur always seeing Tedd in girl form notwithstanding, there is no chance that Arthur might not become aware of Tedd having and using boy form. Hiding this information from him is technically part of not outing Tedd, however it will be short-lived and would mean missing opportunity to control how Arthur finds out Tedd has boy form.
The preferred option here might be insisting Tedd is boy. At worst case, Elliot and Ellen might seem stupid and/or rude to Tedd, but there is reasonable chance Arthur will get the point that Tedd being girl is not public, while being completely in sync with not outing Tedd to Ashley.
(Why it seems perfectly normal to me to use "girl" for 18 year old female but weird to use "boy" for 18 year old male?)
Not true. Sure, if it's bigotry, Arthur could yell all day, but if it's ignorance, Arthur is in position to cause Edward think, because unlike Tedd he's adult and authority figure.
(Wait ... he's adult or he's an adult?)