I think it is a toss up between using a short carbide drill and interpolating.
Through holes can be problematic because as the drill breaks through everything gets very hot; if you do not use coolant it can be hot enough to burn the cutting edge off the drill in addition to throwing up a horrendous burr. If you do use coolant then the intermittent cooling thermally shocks the cutting edge and you get microchipping and early failure.
I would be tempted to try one of the drill/mill cutters that has a drill point on the tip and milling flutes further up. Something like a 1/2" could possibly be punched through fairly quick just using a powerful airblast for cooling, then the same tool interpolates the hole out to 7/8" again with the airblast. A distinct advantage to this approach would be an absolute minimal burr on the underside.