Is there a reduction to "door and pressure plate" games that doesn't explode solution length?
Perhaps you can easily simulate a LBA; the idea is the following:


for every cell of the LBA tape add a cell gadget $G_i$ that can be entered only from the bottom and leaved only from the top;
the gadget has an entrance door $C_i$ which simulates the head position (only one $C_i$ is opened at every step);
then there are two bit doors $Z_i$ and $O_i$; $Z_i$ is opened if the cell contains a  zero, $O_i$ is opened if there cell contains a one;
both bit doors lead to a similar control structure which is made by several one-way corridors; a corridor corresponds to a state of the LBA, and the door $q_i$ of the $i$-th corridor is opened if and only if the current state of the LBA is $q_i$;
according to the (possibly non-deterministic) transition table of the LBA, a traversal of the (opened) corridor changes the current state of the LBA and the configuration of the bit-doors, closes door $C_i$ and opens $C_{i+1}$ or $C_{i-1}$.


A cell gadget is sketched in the figure below.



Non-deterministic choices can be realized splitting the corridors  in the control structures into two or more sub-corridors as shown in the figure below.



Note: if a plate can only open/close a single door, then you can add an auxiliary structure with (long) one way corridors that (de)activates the distinct state doors of each cell.