Here's a different type of CR2032/CR2025/CR2016 battery holder, that has the cell clip into it from the top rather than the side.
The CR2032 is the "go-to" size cell for many wearable projects. It offers 3V, has a decent capacity and is relatively inexpensive. Its disk shape is easily mounted into your project with this: The Bhold2032a battery holder.
Designed originally for surface-mount circuit boards, this holder comes with some big metal pads that are convenient to solder wire to, and the frame slots are particularly handy for sewing it to wearable projects.