If the expansion of the universe is accelerating, doesn't that mean that the entire universe is a non-inertial frame of reference? And if so, what implications does this have (if any)?
No. The expansion of the universe and the accelerated expansion is simply a scale factor change, not any kind of motion at all. For there to be a real acceleration there would have to be a preferred direction to the acceleration. The universe is isotropic so there is no direction for the acceleration, the scale factor change simply increases the distance between any pair of points.