Usual preface, I’m with PreciseBits so while I try to only post general information take everything I say with the understanding that I have a bias.
We sell to a LOT of luthiers that cut MOP/shell. Most of them are either using CA (superglue) or hide glue. Usually it’s directly glued to either tempered masonite or MDF. Although a fair number do use tape and CA.
For CA you should be able to pop the pieces off with acetone or a heat gun if glued directly to something like MDF.
There’s 2 things I can think of that might have you breaking tooling using CA with or without tape. Either the type of CA itself or the tooling.
If you are using a “gel” type or other CA that has additives those can (not always) keep it from properly curing in the middle and/or make it more flexible/“tacky” and that can load the tool. The one brand I’m aware of that is regularly used without issue is BSI’s INSTA-CURE+ (Link). You’ll find a lot of people resell it under another name. It almost always still has BSI’s foil label though.
The other thing I’ve seen that can cause an issue with this is using a burr like a diamond-cut or chip-breaker PCB router. They will work but they are much more likely to load from things like glues or adhesives.
Hope that’s useful. Let me know if there’s something I can help with.