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 image 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 his way of keeping the secret wasn't really "less stressful" than Borden'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.