As others have mentioned, try the fixturing wax. This will also make it easier to remove your thin piece without damage.
If you’d like to try a run with the tape method, try a cut 0.003" below the bottom. I’m not positive this is enough but it should stop before accumulating much tape adhesive. It might be possible to cut deeper then clean your bit when done as long as it’s only the last pass that punches through.
Copper is tough to mill except for C145 (a.k.a. C14500, Tellurium Copper). This mills like a charm but seems impossible to find in sheets. Otherwise copper seems to want to deform more than cut. If brass is suitable for your project it’s (certain alloys) much more amenable to milling.
Edit: BTW, silver mills really well too! But with most metals use the more hardened varieties for milling.
Well you’re in for a treat, the machine and the community are both awesome (my humble opinion)
Another aside just to keep in the back of mind, the nomad is plenty capable of making other items to facilitate your work. Just the other day I accidentally made a really clean punch tool for thin brass stock. If you ever find yourself doing repeat parts, maybe you can mill a punch tool yourself (those little round pieces that go on the back of earrings seem like decent candidates, I don’t know much about jewelry making…)
Don’t fall into the trap of only milling just because all you’ve got is a mill. It’s hard to make another hammer if all you’ve got is a hammer but if all you’ve got is a nomad…the possibilities are much more interesting
Something to keep in mind with copper is its hardness state.
Annealed or soft copper sheet ‘O’ temper will be the gummiest. Be difficult to make chips.
1/4 hard would be less gummy, and 1/2 hard better yet.
Copper work hardens with rolling or hammering. However, heating and then gently cooling it will anneal it and make it softer.
It is also very grabby, so sharp tools. And high speeds steel would be an even better option than carbide.
Not all brass mills equally well. The best is 360 brass but the thinest I found with a cursory search is 1/8" (McMaster-Carr is a good source). A type called “architectural brass” or “architectural bronze (C38500 Brass)” also mills well. The “jewelers brass” may be too soft for clean milling. For example, the brass sheet at Rio Grande is “dead soft” and is plyable for bending but not good for milling.
(those little round pieces that go on the back of earrings seem like decent candidates, I don’t know much about jewelry making…)