Tips on dealing with bulky/irregularly shaped pieces in drill press?

Most of the time I can use the mating part/flange/holes as a drill guide. I use the drill on size of that clear hole to generate my center in the broken hardware. I haven't dealt with the same parts you have though.
There was one case at home where C clamped the parts together with nuts captured between them because the clear hole flange wasn't deemed thick enough to guide my drill. I forget what parts that was.
In my plant maintenance days we could trot down to the shop and whip out spacers, bushings, anything we needed to keep a hand drill aligned.