No, neither would be even somewhat true. Of course it's open to interpretation as all language is, but "so I can't" heavily implies there was nothing more to be said about it, because everything else is apparent from context. With regard to the parent, that's a lie in the sense of obscuring ulterior motives. An apt phrase to describe this is lie by omission. In your sense, it's to weasel out, for lack of a better word, because it implies you are too worn out to even argue - without alleging you had ulterior motives.
Yes just replace "I can't" with "I rather don't want to". I don't know anybody who would imply hidden ulterior motives in this case in real life, since "I can't" is not necessarily as strictly interpreted as you are trying to do.
