An APT Multi-Tool would be the perfect tool for your application.
Specifically designed for drilling large holes with low horsepower machines
lacking rigidity like most drill presses andr mag base drills. I have never
used one for 1 inch thick steel, but I see no reason it would not work.
Basicly it is an interchangeable blade spade drill with a pilot. First drill a
3/4 inch hole with a conventional drill bit (or multiple steps working up to
3/4 inch.) The 3/4 inch pilot on the Multi-Tool is guided by the initial
drilled hole. For bigger holes there are 1.5 and 2 inch interchangeable pilots
so the hole can be done in multiple steps, but you will not need that for your
1 5/16 hole. I have a spare set, but many of the blades can use touching up.
I have been trying to get around to sharpening the blades so I can list it on
ebay, but I have not had my surface grinder fired up for a while.

Mill


I'd like to drill a 1 5/16" (1.3125 inch) hole in a 1 inch thick mild steel
bar. I only have a drill press in my shop. Is there a way to drill a hole
this big in a home shop, without a milling machine?

It has been suggested I try a HSS hole saw at low RPM (less than 400) with
plenty of cutting fluid.

It doesn't have to be pretty, and the jig I'm making doesn't require super
high precision.

Thanks,
Dave