1. Note, a huge bonfire may be so hot that the aluminum screen would have to be quite high up, and quite large, to avoid melting it. A stainless steel screen may be more practical. A modest sized campfire for cooking and warmth should be okay though. (You didn't specify, so, making that clear.)
2. There's probably little charge coming off the fire itself, of either polarity. Charcoal is conductive, and flames contain ions (especially the readily ionized sodium, not even so much by quantity as by sheer optical output) thus making the fire somewhat conductive and keeping charge equalized.
3. You can still do it, by alternating charged screens. You need multiple screens of course, insulated from each other, and charged with high voltage. Screens are okay for the charging grids, but the precipitators should be flat vanes to reduce the air velocity and turbulence flowing over the stuck particles.
It's probably more practical to spray the tarp with water, or spread dirt on it, or just not keep it so close to the fire. Or not place it downwind of the fire, but, there's the old campfire superstition about saying/doing things to make the smoke go away from ones' position that always seems to work never but sorta... Basically, the wind is always changing at ground level, so not much help there.
Tim