When gluing the bridge I first mask the area around the bridge to minimize glue contamination on the top. It's fairly easy to mark the tape with a sharp pencil and then cut out the bridge gluing area by hand with a sharp xacto knife.
The bridge is easily clamped while gluing. A block is placed on the back side of the top to provide a flat surface for clamping.
During the gluing process, you can remove one clamp at a time and clean major glue squeez-out. Once the bridge glue joint is cured, removing the tape leaves a nice clean interface between the bridge and the top.