I don't have a good answer; here are some partial answers:
0) If you never use the concept of relativistic mass, you don't
need to know whether it's a scalar or not. This suits me fine.
1) Certainly it's not a Lorentz scalar.
2) Relativistic mass, as "usually" defined, behaves like the
timelike component of 4-vector. It behaves like energy.
Therefore it is invariant with respect to spacelike rotations,
but not invariant with respect to rotations in the xt plane.
So my guess is that those who take a three-dimensional view of
the world think relativistic mass is a scalar, and those who
take a four-dimensional view think it's not (or, more likely,
don't think about it at all).