[Help] Fixing the bed

I'm currently attempting to attach my headboard and footboard to the rails on my bed. The mounting brackets are laid out correctly, but the rails aren't the right height for the headboard and footboard. Specifically, the holes in the headboard/footboard are 15/16 of an inch higher than the corresponding holes on the rails.

So, the idea that I had was to drill holes partially through 4 blocks of wood (one per foot on the bed), leaving just a hair over 15/16 inch in height. That way, the feet on the bed would be secured within the holes, with no chance of sliding off the blocks, and the bed would be raised to the same height as the headboard/footboard.

The problem with this plan is that the feet of the bed are wide. Specifically, they're 2 3/4 inch across at the base, which is wider than just about any drill bit that I can come up with. The only exception that I can find is a Forstner bit, and the only bits I can find that are wide enough to do the job are intended for drill presses.

So this brings me to you, r/DIY. Can any of you think of a way, in the absence of a drill press, to drill a 2 3/4 inch wide flat-bottomed hole part way through a block of wood? Barring that, can anyone think of another potential way to raise the rails on my bed by just shy of an inch while still maintaining some semblance of stability?

Thanks in advance!