= Solution
Write $S_n(t)=\sum_{r=-n}^na_re^{irt}$. First we prove <decay of coefficients of an almost-everywhere convergent trigonometric series> rather than assuming the series can be integrated term by term. For $n\ge1$, $T_n(t)=S_n(t)-S_{n-1}(t)=a_ne^{int}+a_{-n}e^{-int}$ tends to zero outside the finite set $E$. Suppose $M_n=|a_n|+|a_{-n}|$ does not tend to zero, and choose a subsequence with $M_n\ge\varepsilon>0$. Then $|T_n/M_n|\le1$ and $T_n/M_n\to0$ almost everywhere. The <dominated convergence theorem> gives
$$
\frac1{2\pi}\int_0^{2\pi}\left|\frac{T_n(t)}{M_n}\right|^2dt\to0.
$$
<orthogonality> of the two distinct <Fourier modes> instead makes the integral $(|a_n|^2+|a_{-n}|^2)/M_n^2\ge1/2$, a contradiction. Thus $\boxed{a_n,a_{-n}\to0}$.
Define on the real line the twice-integrated <continuous function>
$$
F(t)=\frac{a_0t^2}{2}-\sum_{r\ne0}\frac{a_r}{r^2}e^{irt}.
$$
The series has both <absolute convergence> and <uniform convergence> because its coefficients are bounded. Its symmetric second quotient is therefore, for nonzero $h$,
$$
\frac{F(t+h)-2F(t)+F(t-h)}{h^2}=a_0+\sum_{r\ne0}a_re^{irt}w(rh/2).
$$
We next prove that this tends to zero whenever $t\notin E$, lifted periodically to the line. Let $b_0=a_0$ and $b_n=a_ne^{int}+a_{-n}e^{-int}$ for $n\ge1$, so the partial sums of the $b_n$ are exactly $S_n(t)\to0$. Put $k=|h|/2$ and $w_n=w(nk)$. <Summation by parts> gives
$$
\sum_{n=0}^\infty b_nw_n=\sum_{n=0}^\infty S_n(t)(w_n-w_{n+1}).
$$
The boundary term vanishes because $S_n(t)$ is bounded and $w_n\to0$ for fixed nonzero $k$. Moreover
$$
\sum_{n\ge0}|w_n-w_{n+1}|\le\int_0^\infty|w'(x)|dx=C<\infty.
$$
Indeed $w'(x)=O(x)$ near zero and $|w'(x)|\le2/x^2+2/x^3$ for $x\ge1$. Each fixed weight difference tends to zero as $k\to0$. Splitting the last series at an index beyond which $|S_n(t)|<\eta$ therefore bounds its upper limit by $C\eta$. Letting $\eta\to0$ proves the claimed zero second quotient. This explicitly proves the needed <Riemann summation of a convergent trigonometric series>, despite the oscillations in its weights.
We also need the elementary fact that a <continuous function> with zero <symmetric second derivative> at every interior point of an interval is affine. For a real-valued function, subtract its endpoint chord on any compact subinterval $[a,b]$. If the difference were positive somewhere, adding $\varepsilon(t-a)(t-b)$ for sufficiently small $\varepsilon>0$ would retain a positive maximum at an interior point. At that maximum each symmetric second difference is nonpositive, while its quotient tends to $2\varepsilon>0$, a contradiction. Thus the function lies below its chord. Apply the argument to its negative to get the reverse inequality. Applying this to real and imaginary parts proves the complex-valued case as well. This establishes that <vanishing symmetric second derivative forces an affine function>. Hence $F$ is an <affine function> on each open interval between the finitely many exceptional points in any period.
It remains to exclude corners at those points. Multiplying the formula for the second quotient by $h$, and using $h=2k$ and the evenness of $w$, gives
$$
\frac{F(t+h)-2F(t)+F(t-h)}h=a_0h+h\sum_{n\ge1}(a_ne^{int}+a_{-n}e^{-int})w(nh/2)\longrightarrow0
$$
for every $t$, by the infinite-series version of part (i), since $a_n,a_{-n}\to0$. <continuity> extends the adjacent affine pieces to each exceptional point, and the fact that <symmetric second differences detect an affine corner> makes their slopes and intercepts agree. Thus $F$ is globally an <affine function> on the real line.
The periodic part of its definition gives $F(t+2\pi)-F(t)=2\pi a_0t+2\pi^2a_0$. An <affine function> has constant increments, so $a_0=0$. The resulting $F$ is both an <affine function> and a <periodic function>, hence constant. Finally, absolute <uniform convergence> justifies integrating its defining series against $e^{-irt}$, and for $r\ne0$ the resulting <Fourier coefficient> is $-a_r/r^2$. A constant has all these coefficients zero. Therefore $\boxed{a_r=0\text{ for every }r\in\mathbb Z}$, proving <uniqueness of a trigonometric series outside a finite set> with all exceptional points accounted for.
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