I don't know.  The following is a near miss which might be useful. 

Start with a hexagonal cycle path ABCDEFA.  Duplicate point C to C' and connect C' to
B,C, and D.  Similarly duplicate points E and F, and add edges EE', FF', and the 3 edges
to form the path DE'F'A.  Then it has diameter 3, but the only point that is distance 3 from E (and also from E') is B, so it cannot accomdate such a permutation.  The only problem is that vertex D has degree 4, so the graph is just shy of being 3-regular.

It may be possible to use this by stitching together two large even cycles to get
a regular graph (with the property that two vertices must share an antipode), but I will let someone else do it.

Gerhard "Cycles Can Make Me Dizzy" Paseman, 2011.05.11