Bev, as you mentioned, the choice of gauge would depend on the weight of the beads used to make a hatpin, as well as the size of their holes.
One of our WS Faculty members collects antique hats and sometimes makes hatpins. She uses a full hard temper wire in gauges from 18 through 20. Because copper is naturally a soft metal, you might try using a rubber hammer and bench block to harden the wire without changing the shape.
Another way to harden round copper wire would be to put it into a drill and twist it. No, you will not see a difference in the shape of the wire, it will still be perfectly round, but it will be harder. Just don't twist too much or it will become brittle and could break or snap more easily.
Answer contributed by Dale "Cougar" Armstrong