I use CA (GluBoost) on mine and tried to research the good, bad and ugly that might be associated with that. I talked with several different turners, contacted GluBoost and I think maybe asked on here too. Essentially the research said that no one knew of a down side for the longevity of the stopper with any level of data supported reasoning.
I wrote it that way on purpose... I had a couple people raise questions on the possible affect of refrigerator cold exposure if the stopper was in a bottle of wine in the fridge and then came out of it to a warmer environment. By that, they didn't know if it would be a concern or not. In the course of conversation with GluBoost (since that's what I use) I brought that concern up and they said there was no reason it couldn't be on a stopper or in that type of temperature change.
I don't do a ton of stoppers but do sell a few a year and have never had one come back for an issue with the finish or anything else. We use a couple here at the house that I've made and they've never had an issue. I somewhat watch/review them like a hawk to monitor for possible issues and haven't found one as of yet. One of my stoppers is a hybrid wood & resin and it's never shown any issues either.
The only issue I've run into is making sure that I don't make too crazy a shape or small detail because it can make it harder to get the CA to lay smooth (user issue, not the finish) and polish up nicely.
I've made a fair number of cigar holders from Craft Supplies this year and use GluBoost on those too. I have the supplies in the shop to make a pepper mill and just haven't pulled the trigger on getting it done yet. Not sure if my finish will be GB or lacquer and will decide when I get there.
I'll be curious to see what others use. I know there are several people on IAP who use spray lacquer or straight poly on bottle stoppers or pepper mills.