Built miniatures for years. My trick for small brads was using a pair of small wire cutters to gently hold the brad while using a small hammer. Turn the cutter upside down with the brad head inside the concave and facing you, push down, then hammer in the rest of the way without the cutter. Don't hold the brad too hard, (can cut it off) just enough to give some pressure and start the brad. If the wood is soft, you can also just use the cutters to insert the brad all the way with pressure.

The other option is skip the brad/staple option and us Fabric TacK. Yup, says Fabric, but it attaches anything and can be bought at hobby/fabric stores. Is clear and sets up within a half hour. Use masking tape to hold the wire in place. Once it is close to dry you can rub off any excess as it is pliable until completely dry. Even if you make a mistake you can rub it off. If the wood is going to be stained, do this before using the glue. I use this stuff for all types of wood/fabric/metal trim.