You can re-enable interrupts in specialized circumstances, but you would need to be pretty damn certain that you won't get into too many recursive calls. And if you are certain that won't happen it probably isn't necessary in the first place. Most of the problems with having interrupts disabled did not have the effects the OP thought they did.
The main reason I could get from the original post was that he wanted to be able to do serial comms for debugging, which is the one thing which would slow down the ISR so much that you might indeed get runaway recursion. So really, redesign is needed. Doing a sei() inside the ISR is not going to be the golden bullet that solves everything.