Brian,
Sorry a little late here for my 2 cents.
We print quite a bit of 10Mil LexEdge II for one customer - around 200 Yards/year.
We cut 100% through the adhesive and another approx. 75% through the LexEdge and then fold back at a straight edge and then forward and peel the parts out. These are larger parts and come out pretty easy when the Env 375 is set right. These parts are around 4.5" by 22". We print 4 spot colors and then laminate the adhesive and then cut/weed.
Depending on which Gerber plotter you have, your results and methods will vary from single cuts to double or more. Small circles are not round on an Envision and but much better on and HS or GS series from my experience.
Finally, your question...
We use 3M 467MP which is a 2mil. or you can also get 3M 486MP which is a 5mil. The 467MP is the industry standard for overlays like yours and metal and plastic nameplates, etc. A VERY GOOD point was made already...the surface that the adhesive will adhere to will determine the adhesive that you want to use. Certain surfaces such as some plastics have a Low Surface Energy or grab and require a more aggressive adhesive, while other surfaces such as clean, oil free stainless, aluminum, gloss painted metals(non-powdered coated) will have a higher surface energy grab to them.
Search ebay for: 3M 467MP 12"
Hope it goes well!