How can I relocate the primary lockset bore in a door?
Your choices with regard to this depend greatly upon the materials, construction and finish of the existing door. 


If it is a solid wood door that will end up with a painted finish then it is possible to clean up the existing hole and make a surface to surface plug of wood that is glued into place before starting the drilling for the new 2 3/4 inch backset.
If it is a door that has a wood laminate inner and outer surface but a particle board core it can be possible to perform similar steps as outlined above for solid wood doors. Truing up a particle board inner core can be harder to get good glue surface, particularly if the original hole was cut with a coarse toothed hole saw that chewed up the inner surface of the hole.
For plastic or steel clad doors the most likely thing you are looking at is to replace the whole door with a new one. I have seen some types of tricks used for doors of this type that add brass plates to both sides of the door that are secured to the existing door surface. The added plates are used to support the new cutout and bridge over the old one.