It would be pretty difficult to track 3-12 satellites at a time with a directional antenna, particularly with both the satellites and the plane in motion.
You could employ something like a directional antenna who's radiation pattern will basically look like a half-circle, with the flat part facing down, but, radiation patterns are not black and white, even if 90% of the energy is received from the main side, there is always some signal still received from unwanted directions. With as weak as GPS signals are, it wouldn't take much to drown them out, even on the back side of a directional antenna.
Yes, that's pretty much how military anti-jamming works. You have a steerable (via phase switching, not physically moving) antenna that can form a beam with nulls pointing at the sources of interference/jammers. See https://en.wikipedia.org/wiki/Zero-forcing_precoding for more info.
You could employ something like a directional antenna who's radiation pattern will basically look like a half-circle, with the flat part facing down, but, radiation patterns are not black and white, even if 90% of the energy is received from the main side, there is always some signal still received from unwanted directions. With as weak as GPS signals are, it wouldn't take much to drown them out, even on the back side of a directional antenna.