= Solution
Interpret $\theta$ as a continuous finite-dimensional complex <group representation>, so its values lie in $\operatorname{GL}_{\mathbb C}(V)$, and interpret approximation in the <uniform norm>. Assume the usual Hausdorff convention for the compact topological <group>. The following argument proves the required form of the <Peter-Weyl theorem> without first assuming that $G$ is a <Lie group>.
Start with normalized <Haar measure>. It is both left- and right-invariant on a compact <group>. Average any <Hermitian inner product> to make a finite-dimensional representation unitary. The left regular action $L_a h(x)=h(a^{-1}x)$ is already unitary on $L^2(G)$.
Choose a continuous nonnegative <function> $k$ supported in a small symmetric identity neighbourhood, with $k(z^{-1})=k(z)$ and $\int k=1$. Such <functions> are obtained from a nonzero continuous bump near the identity, symmetrizing and normalizing. Define
$$
T_kh(x)=\int_Gk(y^{-1}x)h(y)\,dy
=\int_Gk(z)h(xz^{-1})\,dz.
$$
The <convolution> operator $T_k$ commutes with <left translations>. Its continuous kernel is Hermitian, hence it is self-adjoint. It is compact on $L^2(G)$: approximate the continuous kernel uniformly by finite sums of separated <functions> of $x$ and $y$, using the <Stone-Weierstrass theorem> on the compact product. The corresponding operators have finite rank and converge in operator norm. It also maps into $C(G)$ and satisfies
$$
\|T_kh\|_\infty\leq\|k\|_2\|h\|_2,
$$
by the <Cauchy-Schwarz inequality>; continuity follows from the <uniform continuity> of the translated kernel.
The <spectral theorem for compact self-adjoint operators> decomposes the <orthogonal complement> of its kernel into finite-dimensional <eigenspaces> with nonzero <eigenvalues>. Every such <eigenspace> is left-translation invariant, and all its elements are continuous, since $h=T_kh/\lambda$ there. If $W\subset C(G)$ is one of these invariant spaces, write $\rho(a)=L_a|_W$. Evaluation at the identity gives
$$
h(x)=\operatorname{ev}_e\bigl(\rho(x^{-1})h\bigr)
=\bigl(\rho^\vee(x)\operatorname{ev}_e\bigr)(h).
$$
Thus $h$ is a <matrix coefficient> of the <dual representation>. This identifies each finite spectral sum with finite-dimensional representation data.
To get uniform rather than just $L^2$ approximation, let $S_N$ project onto increasing finite sums of the nonzero <eigenspaces>. For $f\in C(G)$, $T_kf$ is orthogonal to the kernel, so $S_NT_kf\to T_kf$ in $L^2$. Applying the displayed smoothing bound gives
$$
\|T_kS_NT_kf-T_k^2f\|_\infty\longrightarrow0.
$$
Meanwhile, choosing the support of $k$ sufficiently small makes $T_kf$ uniformly close to $f$, by <uniform continuity> under small translations. Since $\|T_k\|_{C\to C}\leq1$, it also makes $T_k^2f$ uniformly close to $f$. Every $T_kS_NT_kf$ belongs to a finite sum of invariant <eigenspaces>, hence is a finite sum of <matrix coefficients>. This completes the <convolution proof of uniform Peter-Weyl approximation>.
A coefficient $\ell(\theta(g)v)$ equals $\operatorname{tr}(\alpha\theta(g))$ for the rank-one <endomorphism> $\alpha=v\otimes\ell$. Finite sums of coefficients can be combined using a direct-sum representation and a block-diagonal $\alpha$. Therefore the conclusion has exactly the requested form:
$$
\boxed{\text{For every }\varepsilon>0\text{ there are }V,\theta,\alpha
\text{ with }\sup_{g\in G}|f(g)-\operatorname{tr}(\alpha\theta(g))|<\varepsilon.}
$$
For a <class function> $f$, apply <conjugation averaging on a compact group>,
$$
\mathcal AF(g)=\int_G F(hgh^{-1})\,dh.
$$
It fixes $f$ and is a contraction in the <uniform norm>, so averaging an approximation preserves its error bound. A <trace> coefficient averages to $\operatorname{tr}(\overline\alpha\theta(g))$, where
$$
\overline\alpha=\int_G\theta(h)^{-1}\alpha\theta(h)\,dh.
$$
This <endomorphism> commutes with the representation. A finite-dimensional <unitary representation> splits into irreducibles by repeatedly taking invariant orthogonal complements. On each irreducible diagonal block, <Schur lemma> makes its average a scalar multiple of the identity, namely its <trace> divided by the block dimension. Off-diagonal blocks do not contribute to the total <trace>. Thus the averaged coefficient is a finite <linear combination> of <irreducible characters>. Consequently \b[irreducible complex characters span a uniformly dense subspace of the continuous <class functions>].
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