Another way to make 'JSON' smaller is to instead use 'CBOR', schema compatible 'concise binary object representation'. (See IETF RFC 7049 or http://cbor.io/ ) CBOR encodes and decodes faster too. Or use the 'snappy' compressor.
LZ4 is a LZ77 family algorithm which means its dictionary is a "shifting window". I don't believe such kind of dictionary will fit in this case.