$NP$ completeness of Hamiltonicity of cubic polyhedral plane graphs with bounded face degree?
In the first paper which you link to (Garey-Johnson-Tarjan) they show how to construct a 3-input OR on Figure 6. This gadget has the property that it can link 3 edges with the only requirement that at least one of them needs to be used. So start from your original cubic 3-connected graph and suppose that it has a face larger than $d$. Then insert this gadget into this face onto edges $e,f,g$, where $e$ and $f$ are adjacent, and $g$ is opposite to them. Since $e$ and $f$ are adjacent, anyhow one of them had to be used in the original cubic graph, so the gadget will do nothing. On the other hand, since $g$ is on the opposite side of the face, the size $d$ will be almost halved. Here 'almost' means that this inserted gadget also has some size $c$, so the new faces will have size at most $d/2+c$. Repeating this gets rid of all faces whose size is more than $2c$.