If it really will cost 15 to 20 bucks to farm it out then that is your best bet.
I've had similar problems to solve before and if it was me I would use a carbide bur in die grinder. You're only removing a 1/16 on each side of the hole and a carbide bur at high speed will remove a lot of material in a short amount of time. If you have an eye for detail you can make a surprisingly round hole. It will function in an electric drill but will take vastly longer. The HSS burs sold at the hardware store will not do at all!
The trick is of course that you will have to buy a carbide burr which will cost 10-20 dollars at your local industrial supply house.
The hole saw idea will definately work IF you can make the 3/4 pilot to get it started without walking.
I don't think 200 rpm is slow enough for a 7/8 drill and most likely will try to grab while trying cut such a small amount and shatter the edge of the flutes.
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