= Solution
Both the <muon> and <electron> have the same <electric charge> magnitude. At equal kinetic <energy> $K$ per particle, their <Lorentz factors> are $\gamma=1+K/(mc^2)$, so the heavier <muon> has the smaller <Lorentz factor>. At equal total <relativistic energy> the same ordering follows from $\gamma=E/(mc^2)$, provided that <energy> is large enough to create either particle.
\b[If the equal-<acceleration> comparison of <4/c> is retained], $P_{\mathrm{circular}}=C_qa^2\gamma^4\le C_qa^2\gamma^6=P_{\mathrm{linear}}$. Thus the four-way instantaneous comparison selects
$$
\boxed{\text{a muon in a circular accelerator, at fixed energy and coordinate acceleration}.}
$$
For positive $K$, its smaller <Lorentz factor> reduces the <power> relative to an <electron>, and transverse <acceleration> reduces the <power> relative to longitudinal <acceleration> of the same magnitude. This conclusion follows from the constraint actually stated in <4/c>.
\b[For the customary engineering comparison at equal applied <force>], the result changes. From <4/c>,
$$
P_{\mathrm{linear}}=\frac{q^2F^2}{6\pi\epsilon_0m^2c^3},\qquad
P_{\mathrm{circular}}=\gamma^2P_{\mathrm{linear}}.
$$
Under that comparison the choice is
$$
\boxed{\text{a muon in a linear accelerator, at fixed energy and applied force}.}
$$
Its longitudinal radiative <power> is smaller than an <electron>'s by $(m_e/m_\mu)^2$, while avoiding bending removes the extra $\gamma^2$ factor. At a fixed bend radius $R$, the circular loss is $C_q\gamma^4v^4/R^2$, giving the familiar $m^{-4}$ suppression at fixed ultrarelativistic <energy> and $R$.
\b[An <energy> budget alone does not specify a unique optimization.] The <force>, <acceleration>, bend radius, acceleration time, and length are otherwise unrestricted. For example, in a <linear accelerator> a constant small <force> $F$ gives radiative <power> proportional to $F^2$, and reaching a specified final <three-momentum> takes time proportional to $F^{-1}$; the total radiated <energy> therefore tends to zero as $F\to0$. One cannot infer a unique accelerator choice without an additional constraint. The standard practical preference is the second boxed answer, with its stated comparison. <Muon decay> introduces a separate lifetime constraint if slow acceleration is contemplated, but it was not part of the specified radiative-loss calculation.
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