Suggestions for cut efficiency

I’ve got a large order of coasters that I’m working on (750 pieces) that consist of a .08" deep pocket into which I then v-carve a logo @ .14" deep (.06" deeper than the pocket). On the logo cut, the software removes the material from the missing .08", adding to the time for cut. I’ve tried to cut the logo seperately using a small roughing bit, but the detail pass doesn’t completely cut the logo. I feel like I’m pushing the feeds and speeds already, but need to reduce the time it takes to cut the logo. Any ideas? Any way I can force that step over on the detail cut and make it start at the material surface?