From the book  "Breeding Super Bees" by Steve Taber.

He experimented with grafting eggs and described altering forcepts to lift the eggs from the cell bottom. The forcepts had to have jaws that closed to a stop leaving a .009" gap and nicely rounded edges.  He said the jaws would sometimes slide off  the egg if a slight sideways motion was not added. He said eggs could be re deposited in the upright position in the new location so I am assuming that the "glue" was soft and sticky. Claimed eggs could stand quite low temperatures but had to be kept in high humidity conditions. 

Emergence was 50 to 75% but queen quality was hit and miss with some even being raised as workers. He did not pursue it apparently.
Mechanically the bees could probably manage moving eggs but it is not part of their behavior to do so.

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