We’re talking about possible representations of a rotation in the plane. The usual representation is “angle measure”, e.g. radians, degrees, or “binary radians”. This representation makes it easy to compose rotations (just add the angle measures), and the binary version is an efficient use of bits, but if you want to do anything else, e.g. rotate some arbitrary vector, then you need to do a bunch of transcendental function evaluations, which is computationally expensive.
I am recommending using the Cartesian (x, y) coordinate representation instead. This is often called a “complex number” x + iy on the “complex unit circle”, or you can think of it as (cosine, sine) if you prefer. Composition of rotations in this representation is still straight-forward: (a + ib)(c + id) = (ac – bd) + i(ad + bc).
If you need to compress this down to one number for transmission or storage, you can take the stereographic projection (“half-angle tangent”), (x, y) ↦ y / (x + 1), and optionally also reduce the precision. This saves you at least half the bits vs. storing the pair of coordinates, and is relatively inexpensive to compute (takes one division per point).
You are right that to normalize an arbitrary vector requires taking a square root. Fortunately, this doesn’t need to be done too often.
As an added bonus, this method generalizes much more easily than angle measure to representing higher-dimensional rotations and points on higher-dimensional spheres. For instance, I recommend using cartesian coordinates (or the stereographic projection for compression) instead of latitude and longitude for storing points on the 2-sphere such as geographical places. Many pieces of mapping software are constantly converting back and forth (implicitly or explicitly) between latitude/longitude vs. cartesian coordinates for computing distances, directions, and areas, finding intermediate points, applying 3-dimensional rotations, clipping shapes, and so on. This is slow and complicated vs. just using Cartesian coordinates as the internal representation.
It is usually unnecessary to compute the angle measure corresponding to a rotation unless communicating with a human. Angle measure is just one of several possible representations, and it’s only the most common because it gets taught to every child.