Im not so sure you want to pull air from the bottom of your grill box, to be strong enough to pull the smoke it would also pull the heat and flames away from where you want it.
Perhaps have a metal shop fab up a long narrow funnel like shape to fit just under the grate and vent thru the side, then attach the duct, fan and more ducting. Im thinking something like a NASCAR style exhaust only smaller and in reverse.
Go to the local autoparts and pick up an aux.fan (the type that plugs into the cig lighter)mount it so it blows above the grill but below your face, you need a 12volt pwr supply though. I bought a similar size fan that runs on ac at BigLots, you could probably find one at walmart or walgreens