Finally got tired of lid liners slipping out of position while trying to clamp the pieces together, so I decided to use my 3" pad sander and drill press to create a way of clamping without torque moving things out of position.
I positioned the lid liner in the usual way, marked its location on the lid, applied Nexabond in dots all over the liner, held it in place for a few minutes, then put it on the drill press table that was elevated with a few thick pieces of maple. I cranked the table up until it held the lid and liner firmly together, and checked with a flashlight to be sure the edges were fully down.
After a few minutes, I could have removed the assembly and clamped it at the edges with spring clamps, but that didn't seem necessary. I imagine that this could be made more elaborate, but for a small lid, it seems to have worked just fine.