This can be formulated as a MIP model using a gaggle of binary variables. First, we introduce some parameters. $\tau(u)$ is the index of the transmitter with highest weight for user $u$. I will charitably assume this is unambiguous (no ties for closest transmitter). $\overline{Q}_u$ is an upper bound on the possible values of $q_u$ for user $u$. $C$ is the maximum number of transmitters in any cluster and $L$ is an upper bound on the number of clusters.
Let $x_{t,t'}\in \lbrace 0,1 \rbrace$ be 1 if transmitters $t$ and $t'$ are in the same cluster, 0 if not, for $t\neq t'$. $x_{t,t}$ will be 1 if transmitter $t$ "anchors" a cluster and 0 if not. A cluster has exactly one anchor.
Let $y_{t,u}\in\lbrace 0,1 \rbrace$ be 1 if user $u$ is in a cluster with transmitter $t$ and 0 if not.
Here come the constraints. First, we require that the $x$ variables be symmetric (if $t$ is in the same cluster as $t'$, then $t'$ is in the same cluster as $t$):$$x_{t,t'}=x_{t',t}\quad\forall t\neq t'.$$
We also need to enforce transitivity. If $t$ and $t'$ are in the same cluster, and $t$ and $t''$ are in the same cluster, then $t$ and $t''$ are in the same cluster. So we add the following constraints. $$x_{t,t''} \ge x_{t,t'} + x_{t',t''} - 1\quad \forall t,t',t'' \ni t\neq t' \,\&\, t' \neq t'' \,\&\, t \neq t''.$$
We must force every transmitter to belong to a cluster with an anchor, which requires additional variables $v_{t,t'}\in [0,1]$ where $t\neq t'$. The additional constraints are $$v_{t,t'} \le x_{t,t'} \quad \forall t\neq t'$$ $$v_{t,t'} \le x_{t',t'} \quad \forall t\neq t'$$ and $$x_{t,t} + \sum_{t' \neq t} v_{t,t'} = 1.$$ So if $t$ is an anchor ($x_{t,t}=1$), $v_{t,t'}=0$ for all $t' \neq t$; but if $t$ is not an anchor ($x_{t,t}=0$), then $v_{t,t'} =1$ for exactly one $t'\neq t$, and that transmitter $t'$ must be an anchor ($x_{t',t'} = 1$).
We can limit the number of clusters by limiting the number of anchors: $$\sum_t x_{t,t} \le L.$$ (Change that to an equation if you need exactly $L$ clusters.)
The following constraint enforces cluster capacity limits: $$\sum_{t'=1}^T x_{t,t'} - x_{t,t} \le C - 1\quad \forall t.$$This says that the number of transmitters other than $t$ in the same cluster with $t$ is at most $C-1$.
To avoid having multiple anchors in a single cluster, we enforce the following:$$x_{t,t} + x_{t',t'} + x_{t,t'}\le 2 \quad\forall t < t'.$$
Next, we enforce the requirement that each user belong to the cluster containing the best transmitter:$$y_{\tau(u),u} = 1 \quad\forall u.$$To determine what other transmitters serve $u$, we add the following:$$y_{t,u} = x_{\tau(u),t}\quad\forall u,\forall t\neq \tau(u).$$
The objective remains to maximize $\sum_u q_u$. To linearize $q_u$, we introduce variables $z_{t,u}\in [0, \overline{Q}_u]$ for all transmitters $t$ and users $u$, along with the following constraints:$$z_{t,u} \le \overline{Q}_u y_{t,u} \quad\forall t,u$$$$z_{t,u} \le q_u + \overline{Q}_u(1-y_{t,u})\quad \forall t,u$$and$$z_{t,u}\ge q_u - \overline{Q}_u(1-y_{t,u})\quad \forall t,u.$$
The net effect of these is that$$
z_{t,u}=\begin{cases}
q_{u} & y_{t,u}=1\\
0 & y_{t,u}=0
\end{cases}.$$
Finally, we linearize the definition of $q_u$ via the constraints $$\sum_t w_{t,u} (q_u - z_{t,u})=\sum_t w_{t,u}y_{t,u}\quad\forall u,$$which is the result of multiplying both sides of the definition of $q_u$ by the denominator and then simplifying.
Addendum: There's a bit of inherent (undesired) symmetry in the model. If you take any solution and change the "anchor" transmitter in a cluster to a different member of the same cluster, you get the same results but the solver sees it as a different solution. To avoid that, the following optional constraints can be added:$$x_{t',t'} \le 1 - x_{t,t'} \quad \forall t < t'.$$This forces the lowest index transmitter in each cluster to be the anchor. Whether these speed up the solver or not is an empirical question.
Addendum2: Details of both this and an alternate (and apparently less efficient) MIP model, along with construction and improvement heuristics that seem to work really well, are given in a blog post. The blog post also contains a link to my Java code.
