I actually believe that it can be proven.
Proof: Without loss of generality, we just prove it for the case of rotations in 3D. We start with a definition
Definition: We call $q$ a scalar under a rotation if and only if it transforms under the trivial representation of the rotation group. That is, if $q'=Rq=q$.
Now, suppose that $q\in \mathbb{R}$. Then, it is a rank zero tensor. We may write $q = q v_i w_j \delta^{ij}$. It can be shown (I leave it to you) that under rotations that
$$q= q v_i w_j \delta^{ij} \to q v_{i'} w_{j'} \delta^{i'j'}= q v_i w_j \delta^{ij}=q $$
Hence, we have used the fact that the kronecker delta transforms trivially under rotations to show that if $q\in \mathbb{R}$ is a real number then it transforms as a scalar under rotations.