Given that we say things such as "I'd rather (that) you do it.", I'd expect "I had better (that) you do it." to be possible as well to mean "I would consider/find/have it better that you do it.", analogous to "I would consider/find/have it rather that you do it." (rather being the comparative of the archaic rathe).
However, a quick search would show that this isn't used and is likely ungrammatical. Is there any reason behind this?