That's interesting, because I took the sentence verbatim from a linguistics book (辞書には書かれていないことばの話、仁田義雄). It says うな重 is short for 「うな重を食べた人」. It was compared with 漱石を読んだ (-> 漱石の作品) and 及第点に荷物を持たせよう (-> 及第点を取った人)

So you're saying that this sentence makes no sense unless は is used?