All plastics have some innate polarization, it's a byproduct of the internal stresses that form when they're molded (and why Perspex models of components were used for stress analysis before the days of CAD/CAM software). Even planar windows in fixed-wing cockpits are polarized, so a polarizer on your lens (LP or CP) will always introduce problems.
Realistically, the only way to get pro quality aerials is to remove the door, or mount the camera outside on a remote gimbal. With a wide angle lens and a cooperative pilot you *might* be able to get away with a fixed mount on the skid (I'm talking GoPro-style gear, not a 5D) - but they may not accept 'amateur' stuff clamped to the airframe. Without vibration isolation a DSLR can be shaken to bits.
From inside the canopy, you have to accept that you're shooting through a hideous optical surface (even a brand-new bubble is covered in bumps and scratches, they're only meant to keep the rain out). You can't prevent image softening and loss of contrast, but you can reduce the internal reflections in the same way you'd shoot in an aquarium - take a black cloth and hide underneath it, making yourself a lightproof tent against the inside of the canopy. Try to keep the lens a little way back from the inside surface so any imperfections in the line of sight are comparatively small, and avoid the smallest apertures so it's never in focus. If the sun's hitting your side it'll still create flares, nothing you can do to stop that. It's a good idea to wear gloves, so when you rest your fingers on the window to steady yourself you don't make fingerprints.