Lion's lemma is equivalent to several other properties that have simpler proofs, see On a lemma of Jacques-Louis Lions and its relation to other fundamental results:
Some of these equivalent properties can be given an independent, i.e., “direct”, proof, such as for instance the constructive proof by M.E. Bogovskii of the surjectivity of the divergence operator. Therefore, the proof of any one of such properties provides, by way of the equivalence theorem, a means of proving J.L. Lions lemma, the known “direct” proofs of which for a general domain are notoriously difficult.
Neither Bogovskii's proof nor the equivalence proof makes use of Fourier analysis, so in that sense this might be what the OP is searching for.