Why does Robert Angier not use a double for "The Real Transported Man"?

In addition to iandotkelly's already good answer, I'd say it just doesn't fit to Angier's nature. One of the major differences between Borden and Angier is, that Angier couldn't ever imagine to devote his entire life to "just a magic trick", and always wanted to go the easy way, whereas Borden took his art as serious as to only live half a life for his entire life, which in fact was "his real trick and shows true dedication to his art".

That is the reason why Angier couldn't imagine Borden to use a double, since his own fast experiences with this were rather unfruitful (yet, he was lucky enough to find his exact double in London, but nevermind ;-)) and why he couldn't get behind the old Chinese magician's trick, while Borden undertsood it immediately. And this is also the reason why he doesn't want to use a double. As you say, both Angiers would have the same personality and as iandotkelly says, there wouldn't be any one who had more right to be the "real" Angier. So in order to not share his whole life with another Angier to keep the trick secret, he went the fast and easy (and pretty dirty) way of a forced move operation instead of a copy. On the longer perspective Borden's way of keeping the secret wasn't really "less stressful" than Angier's if you consider what Borden had to sacrifice in his non-stage life in order to keep up his trick.

And an additional factor is also, that I for myself think that Angier didn't just do those tricks to do a good show and win against Borden success-wise, but also in order for the supposed "murder" to happen one day and to ultimately win against Borden. And for this one Angier had to die.

I think the problem as Angier sees it is 'which one had the right to get his life'. If he were to allow the other Angier to live, would it end up killing him to replace him. Just because someone has the same personality as you doesn't in my opinion make it less likely that they would feel threatened by you, actually with shared memories and understanding, your doppelgänger might be more dangerous than another person.

I think Angier decides that it doesn't particularly matter which one of them survives, as long as there is only ever one. He even prepares for this in his first experiment with the device, by having the gun available.

Its kind of twisted logic, because for each clone created by the machine, one Angier dies - but he doesn't know which one he is going to be each time. But I think he believes only having one survivor is cleaner .... no chance of "going rogue" happening, or his secret being found out.

Finally, its a dramatic demonstration to himself (and ultimately Borden) that he's finally willing to "get his hands dirty" to perform the ultimate illusion.