Cannot attend CS conference but coauthor isn't registering

Another possibility is that your "non-academic employment" obviously values your skills - have you considered asking them if they will give you leave (paid or unpaid) to let you present - and it has been known that some people say in their intro "I am here thanks to the grace of X...".

The publication is critical as it is the only proof of me producing anything in quite a long time, so I cannot afford to lose this publication

Then you should be turning up yourself. You can't blame this on anyone else: you messed up your schedule.

How do I ensure that someone registers for my paper so that it appears in the proceedings?

Do it yourself. Sorry, it's that simple. You need to start taking responsibility for this now.

I understand that, having left it so late, it's going to be a little awkward getting time off from your new employer. However, presenting an academic project at a conference is a pretty good reason to have that time off — an employer not understanding of that is probably not somebody I'd want to work with. It's just that late notice that you're going to have to contend with. Given its importance to you, you should have had this event as a fixed date on your diary for months now, and noted it to your new employer early on.

It has been discussed at length that you should reconsider if the paper or not leaving to present it are more important and I urge you to do that.

However, assuming you know your situation better than us, even though it sounds off, here's the only solution I can think of:

Ask. Nicely!

Call your co-author, say please (because you want something from him) and hope he says he'll register. Then explain how important it is and ask for a confirmation so that you can sleep better. Make sure it doesn't come across as mistrust and that you're just asking because it's so important to you.

If that doesn't work, swallow your pride and beg. Offer to pay the flight (if you were going you'd pay it too (?)). Offer something else. Most people will agree to do it when they understand the severity of the situation. But maybe try to explain it more in depth than you did here, because I haven't understood it yet.