underbody advice

I have my whole car stripped down and ready to begin body. (probably next spring now) The car is very rust free and any little surface rust that started is getting fixed.
the inderbody is still very intact and the original coating is in good condition.
the rest of the car will be all replaced or repainted, but i think leaving the underbody alone may be best for the car? MY only concern is it will be the ugliest part of the car after it is finished.

So shoould I leave it be?
OR clean it up but not remove it and put a fresh coat over it?
OR try to strip it all down and replace it?

Thanks!