Another way to see unboundedness of the sum is to notice that for 2-periodic $f(t) = \{t/2\}-1/2$ we have $|\hat{f}(m)| \gtrsim \frac{1}{m}$ for $m\neq 0$. Since the series $$\sum_{m \neq 0} \left| \frac{\hat{f}(m)}{e^{i\pi mm x}-1}\right|^{2} \gtrsim \sum_{m \neq 0, \mathrm{dist}(mx, 2\mathbb{Z})<C/m} 1 = \infty $$ whenever $x$ is irrational, the claim follows from this proposition

Another way to see unboundedness of the sum is to notice that for 2-periodic $f(t) = \{t/2\}-1/2$ we have $|\hat{f}(m)| \gtrsim \frac{1}{|m|}$ for $m\neq 0$. Since the series $$\sum_{m \neq 0} \left| \frac{\hat{f}(m)}{e^{i\pi m x}-1}\right|^{2} \gtrsim \sum_{m \neq 0, \mathrm{dist}(mx, 2\mathbb{Z})<C/m} 1 = \infty $$ whenever $x$ is irrational, the claim follows from this proposition