Can the battery be replaced in my GW1?

Yes, they are replaceable. We do not supply replacement batteries, but they are available from online suppliers.
Search – PD 3048 Rechargeable Lithium Ion Battery Coin Cell 300mah(Brand – Route JD)
The replacement may be better done by a trained professional (Watch Repairer) so they can check the unit is sealed correctly after swapping over battery to stop possible condensation issues and water getting into the unit.