= Solution
\b[Metric-sign qualification.] With this paper's $(-,+,+,+)$ <metric signature>, <4/b> gives
$$
F^\alpha F_\alpha=\gamma^2\left(|\mathbf F|^2-\frac{(\mathbf F\cdot\mathbf v)^2}{c^2}\right)\ge0.
$$
The minus sign in the printed <Larmor formula> would therefore make a positive radiative loss negative. A concrete counterexample is an instant of rest with nonzero <acceleration>: $F^\alpha=(0,m\mathbf a)$, so the printed expression is $-q^2a^2/(6\pi\epsilon_0c^3)$. That sign is appropriate for the opposite <metric signature>. With the convention already fixed in question 1, positive emitted <power> instead is
$$
P_{\mathrm{rad}}=\frac{q^2}{6\pi\epsilon_0m^2c^3}F^\alpha F_\alpha.
$$
Alternatively the printed negative quantity can be interpreted as the particle's signed radiative <energy> change, rather than the positive loss magnitude.
For fixed <rest mass>, resolve the <acceleration> parallel and perpendicular to the <velocity>, as in <4/b>. This gives
$$
\begin{aligned}
\frac{F^\alpha F_\alpha}{m^2}
&=\gamma^2\left(\gamma^6a_\parallel^2+\gamma^2a_\perp^2-\gamma^6\frac{v^2}{c^2}a_\parallel^2\right)\\
&=\gamma^6a_\parallel^2+\gamma^4a_\perp^2.
\end{aligned}
$$
Writing $C_q=q^2/(6\pi\epsilon_0c^3)$, the <Liénard radiated-power formula> becomes $P_{\mathrm{rad}}=C_q(\gamma^6a_\parallel^2+\gamma^4a_\perp^2)$. For uniform <circular motion>, the <acceleration> is perpendicular to the <velocity>; for straight-line motion it is parallel. Comparing the same particle at equal <relativistic energy> and equal laboratory <acceleration> magnitude $a$ therefore gives
$$
\boxed{P_{\mathrm{circular}}=C_q\gamma^4a^2,\qquad
P_{\mathrm{linear}}=C_q\gamma^6a^2,\qquad
\frac{P_{\mathrm{circular}}}{P_{\mathrm{linear}}}=\gamma^{-2}.}
$$
This particular comparison makes the linear loss larger. The familiar larger loss from bending applies to a different comparison: at equal applied laboratory <force> magnitude $F$, $a_\perp=F/(m\gamma)$ and $a_\parallel=F/(m\gamma^3)$, giving $P_{\mathrm{circular}}=C_q\gamma^2F^2/m^2$ and $P_{\mathrm{linear}}=C_qF^2/m^2$. Stating which quantity is held fixed is essential.
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