Is “I don’t suppose you could lend me a pound?” a well-used expression when you ask someone to lend money?

Yes, "I don't suppose you could...?" is a common, self-deprecatory preface to a request like this. In one of the 'Billy Bunter' stories, the answer is "You must have been oiling your supposer: it's working perfectly!" [i.e. "No, I couldn't/ won't lend you money."]

PS Not a verbatim quote; it's been a long time since I studied Frank Richards' oeuvre.

Let's not get too bogged down with the specific sentence here. The main issue is about starting a request with "I don't suppose..." Googling in quotes gets 127K hits, of which maybe 15-25% are polite/hesitant requests. Obviously the usage is alive and well, though I feel it is just a little dated and mock-deferential.

Interpreted literally, it's a helpless and bereft supplicant with no hope, save that his intended benefactor might contradict him and actually do whatever the supplicant assumes they won't.

The "...you could..." part just amplifies the "deference" by introducing the conditional verb tense as used in the alternative Would/Could you lend me...?" (see @Dusty's answer here). I'm seeing a lot of Google hits for quotated "I don't suppose I/you/we could", and most of them seem to be requests, not statements of belief. Again, the usage does occur quite often.

TL;DR: It's a well-used expression. Right up to the bit about lending. Well-brought-up Anglophones don't tap other people for loans like this.

Yes, "I don't suppose you could..." is a somewhat common way of asking someone to do something. It's used to avoid sounding like you are expecting or demanding that they do the thing, or if you are embarrassed to have to ask them to do the thing.

It is roundabout and disingenuous. You've noted an important thing in that the character has never paid back those loans. It's kind of a verbal equivalent to a dog's tail tucked between his legs. (If that image crosses cultural boundaries.)