Okay guys, I need some help with this. Over the weekend I got my little HF drill press set up and tried to remove the long rivets from the rear trunion of a Romy G kit. On one I drilled all the way through. The drill wasn't exactly straight and I did cut into the part a little bit (I don't imagine enough to hurt anything...) but having drilled all the way through it I'm no closer to having the remaining rivet material out!

The other long rivet, I only drilled halfway down and tried to punch it out. Again, no luck.

How do you remove these?

Part of the challenge is that this oddball-shape is hard to put in a vise and seemingly not strong enough to take a huge amount of pressure in a vise.

You are on the right track.  Take your time and drill them straight (start with a center/spotting drill or centerpunch).  After you drill through, punch out the remainder. If it doesn't punch, chase the hole with a 3/16 bit.  Many of the rear rivets in the kits are 3/16, so you might have to do this anyway.

You might look at using wood or making a metal jig to hold it in the vise.  It seems to help considerably.

SRM

With some rivets, especial the ones with a swell neck, I find it helps , once you have widened the hole in the rivet with a larger sized  drll bit, to carefully punch the edges of the rivet over towards the center of the rivet, and then try to punch the whole thing out. The little bit of rivet from the swell neck part can make it a bit harder to remove. It doesnt always happen, but sometimes it helps.

Quoted: With some rivets, especial the ones with a swell neck, I find it helps , once you have widened the hole in the rivet with a larger sized  drll bit, to carefully punch the edges of the rivet over towards the center of the rivet, and then try to punch the whole thing out.

+1.  The rivet edges that you bend to the center also give your punch something to push against.

After drilling, set a 1/4" nut on the anvil part of your vise.  Wear some gloves, position the trunnion on the nut so it's supported fairly level, hold the punch on the rivet with the same hand that's holding it on the nut, whack the punch with a hammer.  Trying to hold it in a vise is a pain in the butt.  I'm surprised no one has fashioned a little jig to hold it, but the nut seems to work for me.  The hole in the nut gives the rivet somewhere to go and the nut itself gives you enough surface to hold up that little corner of the trunnion.  Once you have it moving, I set it up on the vise with the jaws closed just wider than the rivet and pound it through.  I first use a punch that broke and there's just a short stub left.  It's a little more solid for that first whack.  Then I use a punch narrow enough to go through the trunnion hole.

Don't worry about cutting into the trunnion a little, by the way.