It sounds like part of the mechanism that operates the door latch has broken or melted or fallen off or in some other way failed. I’m not sure about the specifics of the rear doors on a 2006 Yaris, but the solution is to get inside the door and operate the latch mechanically to open the door. But that can be easier said than done.

If you can lower the window in that door, you might find a locksmith or specialist safe-breaker who can use a hook to manipulate the latch. But in many cases, the only solution is to remove the interior door trim to get at the broken mechanism. The problem there, of course, is that the door trim will be hard up against the door jamb and the rubber deals, so removing it can be a huge problem.

In some cases, you’ll be able to remove/lift one corner of the door trim to gain access, but I can’t guarantee that the trim won’t be damaged in this operation. Once it’s damaged, of course, all bets are off and you can cut a hole in the trim to gain the access you need to trigger the latch.

And the better news is that there are plenty of Yarises in wrecking yards waiting to give up their door trim to replace the one you’re about to butcher. And if memory serves, all Yarises had the same colour interior, so finding a matching door trim won’t be too difficult.

It's always a good idea to replace a 10-year-old car to improve safety and economy but recalls for minor things are not a huge worry.

The car with the lower kilometres can be an attractive proposition, but it's a Cruze and should be avoided as they have a terrible history of unreliability. The Yaris is smaller and it has done more kilometres, but it would seem to be a much better bet.