Are there any Drill Bits with thicker bases?

[Paul Perry (Frostwave)]

Would you suggest running the Dremel at the slowest speed?

My experience with carbide drills is limited (I broke them all long ago!) but, I would think yes, slow as you can. Plus, as little pressure as you can.
In actual automated use, they run at such a speed that sometimes they are driled into blocks of aluminum to COOL them!! during use (unlike PCB material, aluminum is a great heat sink & very soft) , but that is a very different world from my benchtop..