The wiki page for rumbling says this: "Sending certain
Pokemon to certain areas does not affect the probability of finding
certain items." So it seems the answer is nope. Kind of a shame
though, the type having an effect on stuff found might add an
interesting bonus layer of strategy to it. :,]
It's a shame it doesn't seem like it matters whether I put an Entei
into a Rumble or a Bidoof (hoping that's a good comparison). Maybe
once the rumble battles get implemented, the specific pokemon will
matter.