Regenerating fossils on Pokemon heart gold?

Say I want an Omanyte with a certain nature and ability, and I don't want to have to breed it endlesly to get the right match, can I take the helix fossil to where it gets revived and save outside it so that I can receive the Omanyte, check if I want it or not, then power off and try again? Or is its nature/ability etc predetermined the moment you find the fossil?