But, but... If the structure was put in place now, before the dog became family, the dog wouldn't view it as temporary, would it??
You're right, so long as it's solid, though some dogs are more prone to escaping than others.
BUT
Ddog isn't much of an escape artist, but he knows where there have previously been holes in DF's garden, which have since been very badly bodged by DF. Result is that he will attempt to go though a bodge job there whereas he would probably ignore it if he had never known a hole there. He climbed a tree and jumped over on one occasion, and simply punched his way through on another.
DF has remarkably nice neighbours, but this has resulted in a situation where ddog has gone through at least 5 other gardens all with equally crap fences and along the road before reappearing in record time, looking very pleased with himself, on DF's front door. 
DF doesn't want to replace his fence, so he's now always on lead in the garden there which is a PITA when it's dark and raining. Ddog is always off lead in the park and has great recall but finds the prospect of holes in fences far too appealing for his own good.
He also checks my garden gate every time to see if it's bolted - it's a temporary structure in his mind (sometimes open, sometimes shut) so he checks it in a way he doesn't with the brick wall.
So, I can also see the rescue's point of view.
A temporary structure might be fine now, but I'd be worried if it was still as sturdy in a year. It only takes one escape for a dog to get run over.
I'd just get your garden fenced and be done with it. It's SO much more convenient just being able to open the back door for early morning / late night wees.