[loicd]

There are weaker forms, those accepted in intuitionistic logic. The law of excluded middle usually appears in mathematical proofs in the form of reasoning by contradiction:

  To prove A, assume not-A and reach a contradiction

This is the non-constructible reasoning par excellence, but the following weaker form is constructible:

  To prove not-A, assume A and reach a contradiction

Intuitionistic logic also allows so-called Ex Falso:

  To prove A, reach a contradiction (without assuming not-A)

Now, if what you wanted is a form of LEM that is non-constructible but 'less' non-constructible than LEM, then I don't know.