= Solution
Assume $\beta$ exists and choose a reduced residue class $a$ with $\chi_1(a)=-1$; such a class exists because $\chi_1$ is nonprincipal. Fix a sufficiently large constant $A=A(\epsilon)$ and take
$$
x=\exp(A(\log q)^2).
$$
Then $x\geq q^2$ for large $q$. Multiplying the formula from part (d) by $\varphi(q)/x$ gives
$$
\frac{\varphi(q)}x\psi(x;q,a)
=1+\frac{x^{\beta-1}}{\beta}
+O\left(
\varphi(q)(\log q)^2
\exp\left[-\frac{cA\log q}{1+\sqrt A}\right]
\right).
$$
Choose $A$ so large that the error is at most $\epsilon/4$ for all sufficiently large $q$. The assumed upper bound then implies
$$
\frac{x^{\beta-1}}{\beta}\leq1-\frac{3\epsilon}4.
$$
Since $\beta<1$, this gives
$$
x^{\beta-1}\leq1-\frac{3\epsilon}4.
$$
Taking logarithms,
$$
(1-\beta)A(\log q)^2
\geq-\log\left(1-\frac{3\epsilon}4\right).
$$
Therefore, with a positive constant depending only on $\epsilon$,
$$
\boxed{\beta\leq1-\frac{c}{(\log q)^2}.}
$$
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