Very nice! Thanks for the idea.
What do the lids look like? Are they the same material as the wood you tested on? Various posters have pointed out the the laser’s quality changes quite a bit based on the material. Woods with very obvious grain patterns usually don’t work well, because the darker grain burns more than the lighter parts.
Double sided tape is probably enough for lasering. Several people have had success using it, and have posted so on the forums, although a quick search only turned up one example.
If you have reservations about it, or try it and it doesn’t work, other users have made lasering jigs to hold their work in place. There have been some clever jigs and clamping system that clamp from the side rather than the top of the work piece. Although some of my lids have silicon baffles on the side of the lid, and those might be harder to clamp from the sides.