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