Glad you found the issue. Wether or not the stator is fried I'd just plug it in and find out. What will kill the stator is if it got hot enough to break down the varnish that insolates the windings. The reg/rect is not as tough and it would be easy for me to believe that it's toast. You know all electrical parts have magic smoke inside and if you let that smoke out, they don't work anymore.
The choice of a battery or a cap. For your aplication, since you don't have blinkers that you want to still blink at idle, I'd go with the cap. Mostly because it's not as heavy.
Good catch finding the short.
XR650R Plated Supermoto
XR628R Plated and highly modified
CR500R Still rips after all these beers
RZ350L Kenny Roberts Special
BMW 1150GS It's like ridding a Barkalounger