bounded outdegree bipartite spanners
Perhaps this is a possible weird reduction from 3SAT. The idea is to use the set $S$ with restricted degree to simulate a true/false assignment to the variables using their odd/even distance from the clauses that contain them.

Given a 3CNF formula with $n$ variables; let $c$ be the maximum among the number of occurrences of the variables in the clauses, and let $k = c+2$.

Build $G$ in the following way:


add a source node $s^T$ linked with $k+1$ nodes of degree 1 (a "fan");
for every variable $x_i$ add a triangle made of three nodes $\{x_i^S,x_i^+,x_i^-\}$ and edges $(x_i^S,x_i^+)$ (left side), $(x_i^S,x_i^-)$ (base), $(x_i^+,x_i^-)$ (right side);
add edges from $s^T$ to every $x_i^S$;
add $k-2$ fan nodes to $x_i^S$;
for every clause $C_j$ add a node $C_j^T$ linked with $k+1$ fan nodes of degree 1;
if positive variable $x_k \in C_j$ then add a direct link from $C_j^T$ to $x_k^-$; if negative variable $\bar{x}_k \in C_j$ then add a link from $C^T_j$ to $x_k^-$ with an intermediate node $z_{jk}$


The resulting graph $G$ has a connected, bipartite subgraph ($S\cup T=V$) in which the nodes of only one of the two set (suppose $S$) has max dregree $k$ if and only if the original 3CNF formula is satisfiable.

First of all the fan of the source node $s^T$ forces it to be in the unrestricted set $T$; and the fan of every clause $C_j^T$ forces it to be in $T$, too. Furthermore all nodes $x_i^S$ connected to $s^T$ are forced in the set $S$.

Now, for every variable, only one among $x_i^+, x_i^-$ can be put in $T$ (traverse the base of the triangle or pass through its vertice) because $x_i^S$ is in $S$ and its $k-2$ fan nodes must be connected, so we must drop edge $(x_i^S,x_i^+)$ or edge $(x_i^S,x_i^-)$). Suppose we choose $x_i^+$, and drop the edge $(x_i^S,x_i^-)$; hence $x_i^-$ falls in $S$, and we can use the edges from it
to safely connect (satisfy) all the clauses (all because $k=c+2$) that contain $x_i$ and are directly connected to $x_i^-$, but not the clauses that are connected through $z_{ij}$ (which must be in $T$) because in that case $C_j^T$ would fall in $S$. Similarly, if we choose to put $x_i^-$ in $T$ then we can conect all the clauses that contain $\bar{x}_i$ (through $z_{ij}$ which now is in $S$). 

At the end the graph $G$ is connected only if every clause contains at least one variable made true by the assignment.