Bullet Points could skip to the future, or...
...Tony got Thor's DNA from the Avengers' very first meeting, right? Which means, over the years, he could have collected dozens of material suitable for cloning - including Cap's. After the final battle, during which Cap might die/fake his death (either way, both the public and the anti-reg side think he's gone, maybe some of the pro-reg, as well), Tony realises how discouraged everone is and releases his clone of Cap, claiming that he has realised the error of his ways and wishes to atone for the mistakes by donning that suit of armour.
It's a possibility, isn't it?