As per the provided screenshot, the door is part of a collection of geometry in the scene using CSG Meshes:
Constructive Solid Geometry are meshes which can add or subtract from each other, but this is a calculation that has to be performed by the system, thus their unique feature may also severely degrade performance if modified at runtime.
In this case, as the door opens and closes it is intersecting the other CSG wall meshes and will be recalculating their combined geometry result on single frame, which resulted in the massive framerate drop as was confirmed.
This was solved by making the Door a regular MeshInstance.
The solution to avoid this is to replace any CSG components that you are likely to animate (or which may freely intersect other CSG geometry) with MeshInstances.
While CSGs have their uses, they are unfortunately somewhat deceptively advertised as being the solution for level prototyping, when in almost all cases MeshInstances will do an equivalent job for basic level prototyping.
Anyway — while MxtApps has solved the problem, I hope that it helps anyone else!