There is an even bigger reduction that can be done:

**Theorem:** The Flint Hills series converges if and only if the series
$$
\sum_{n = 1}^\infty \frac{1}{q_n^3 (q_n\pi - p_n)^2}
\qquad{(1)}
$$
converges, where $(p_n/q_n)_1^\infty$ is the sequence of convergents of $\pi$.

*Proof:* Let
$$
S = \sum_{q = 1}^\infty \frac{1}{q^3 (q\pi - p)^2},
\qquad{(2)}
$$
where $p\in\mathbb N$ is chosen to minimize $|q\pi - p|$. As Wadim Zudilin argued, the Flint Hills series converges if and only if $S$ converges. Now consider the unimodular lattice $\Lambda = \{(q,q\pi - p) : p,q\in\mathbb Z\}$. We can rewrite $S$ as
$$
S = \sum_{\substack{(q,r)\in\Lambda^* \\ q > 0 \\ -1/2 < r < 1/2}} \frac{1}{q^3 |r|^2}\cdot
$$
Here $\Lambda^* = \Lambda\setminus\{\mathbf 0\}$. Next, using the identity
$$
\frac{1}{q^3 r^2} = \int_{s > q} \int_{t > r} \frac{\partial}{\partial s}\frac{\partial}{\partial t}\frac{1}{s^3 t^2} \;dt\;ds
= \int_{s > 1} \int_{t > 0} \frac{\partial}{\partial s}\frac{\partial}{\partial t}\frac{1}{s^3 t^2} [s > q][t > r] \;dt\;ds
$$
we get
$$
S = \int_{s > 1} \int_{t > 0} \frac{\partial}{\partial s}\frac{\partial}{\partial t}\frac{1}{s^3 t^2} \sum_{\substack{(q,r)\in\Lambda^* \\ q > 0 \\ -1/2 < r < 1/2}} [s > q][t > |r|] \;dt\;ds\\
= \int_{s > 1} \int_{t > 0} \frac{\partial}{\partial s}\frac{\partial}{\partial t}\frac{1}{s^3 t^2} \#\big\{(q,r)\in\Lambda^* : 0 < q < s,\; |r| < \min(t,1/2)\big\} \;dt\;ds.
\qquad{(3)}
$$
We can bound the integrand in two different ways, depending on whether or not
$$
N_{s,t} := \#\big\{(q,r)\in\Lambda^* : 0 < q < s,\; |r| < \min(t,1/2)\big\} \leq \max(0,3st - 1/2).
\qquad{(4)}
$$
If (4) holds, then it can be used to bound the entire integral; I leave it to the reader to verify that the resulting integral converges. So let us consider the cases where (4) fails.

Fix $s > 1$ and $t > 0$, and let $D_{s,t} = (-s,s)\times(-t,t)$. If $D_{s,t}$ contains two linearly independent elements of $\Lambda$, then $D$ contains a fundamental domain for $\Lambda$, say $F$; we have
$$
1 + 2N_{s,t} = \#(\Lambda\cap D_{s,t}) = \sum_{\mathbf x\in\Lambda\cap D_{s,t}} m(\mathbf x + F)
= m\left(\bigcup_{\mathbf x\in\Lambda\cap D_{s,t}}(\mathbf x + F)\right)
\leq m(2D_{s,t}) = 4st,
$$
which implies that (4) holds. Similarly, if $D_{s,t}\cap\Lambda = \{\mathbf 0\}$, then (4) holds.

So if we assume that (4) fails for some pair $(s,t)$, then we have $D_{s,t}\cap\Lambda = D_{s,t}\cap \mathbb Z\mathbf x$ for some $\mathbf x = (q,r)\in D_{s,t}\cap\Lambda^*$. It follows that
$$
\max(1,3st - 1/2) \leq N_{s,t} = \left\lfloor \min\left(\frac sq,\frac t{|r|}\right)\right\rfloor \leq \min\left(\frac sq,\frac t{|r|}\right)
$$
and thus
$$
\frac{st}{q|r|} \geq \min\left(\frac sq,\frac t{|r|}\right)^2 \geq \max(1,3st - 1/2)^2 \geq \max(1,3st - 1/2) \geq 2st,
$$
so $q|r| = q|q\pi - p| \leq 1/2$. A well-known theorem now implies that $p/q$ is a convergent of $\pi$, i.e. $(q,p) = (q_n,p_n)$ for some $n\in\mathbb N$. So if we let
$$
\Lambda_c = \{(k q_n, k(q_n\pi - p_n)) : n,k\in\mathbb N\}
$$
then
$$
N_{s,t} = \#(\Lambda_c\cap D_{s,t}).
$$
In other words, the only points which are contributing to the integrand of (3) are points which come from $\Lambda_c$. Reversing the argument of (3) now gives
$$
S \leq C + \sum_{\substack{(q,r)\in\Lambda_c \\ q > 0 \\ -1/2 < r < 1/2}} \frac{1}{q^3 |r|^2},
$$
where $C < \infty$ is a constant describing an upper bound on the contribution to the integral (3) of pairs $(s,t)$ satisfying (4). Thus,
$$
S \leq C + \sum_{n = 1}^\infty \sum_{k = 1}^\infty \frac{1}{(k q_n)^3 (k(q_n \pi - p_n))^2}
= C+\zeta(5)\sum_{n = 1}^\infty \frac{1}{q_n^3 (q_n \pi - p_n)^2}\cdot
$$
It follows that (1) converges if and only if (2) converges.

If this proof was too technical to follow, I'll try to summarize the main ideas: First of all, any rational number $p/q$ which is not a convergent of $\pi$ must satisfy $q|q\pi - p| > 1/2$ (this is a well-known fact). By itself this fact isn't enough to guarantee that the terms coming from non-convergents won't make the series (2) diverge, since you end up comparing it with the harmonic series, which (just barely) diverges. But that's just the crudest possible bound: *most* rationals $p/q$ will satisfy $q|q\pi - p| \gg 1$. Since (2) involves a summation over *all* $q$, there will be a lot of "averaging", and so the "spikes" which occur when $q|q\pi - p|$ is small will be washed out in the long run. In order to formalize this you need to talk about lattices and fundamental domains - basically, the idea is that the number of intersection points of a lattice with a convex centrally symmetric region is about the same as the area of the region except for certain exceptional cases; these exceptional cases turn out to correspond to the convergents of $\pi$.

**Corollary:**
If the exponent of irrationality of $\pi$ is strictly less than $5/2$, then the Flint Hills series converges.
*Proof:* If $\mu(\pi) < 5/2$, then there exists $\varepsilon > 0$ such that for all but finitely many $n$, we have
$$
|q_n \pi - p_n| \geq q_n^{-3/2 + \varepsilon}.
$$
This gives the following upper bound for (1):
$$
\sum_{n = 1}^\infty \frac{1}{q_n^3 (q_n^{-3/2 + \varepsilon})^2} = \sum_{n = 1}^\infty \frac{1}{q_n^{2\varepsilon}}\cdot
$$
But since the sequence $(q_n)_1^\infty$ must grow at least exponentially fast, this series converges.

`sum 1/( ProductLog[e^n]^3 sin^2 n), from n=1 to 1000`

written in Wolfram Language $\endgroup$ – user142929 Mar 13 '20 at 23:16