I think I didn't make myself clear. Regarding the baselined CLL 'grammar', which is a pseudosyntax, you can either (a) reject it as irrelevant junk (-- the move I would favour) or (b) treat it as an actual syntax. If you go for (b) then the members of a syntactic category must have the same behaviour with respect to the rules that translate into logical form, and hence if {tu'a} is in LAhE syntactically then (by definition) it is in LAhE semantically.
Actually, (a) and (b) presuppose that we are describing a language, but there's also option (c), which is to describe something that isn't actually a language but nevertheless involves a set of rules mapping pseudosyntax to logical form. I guess (c) is what you're actually doing, which as an intellectual exercise is fair enough.