1000 feet of climbing is hardly anything if you live in a hilly area. Even my "flat" training route has more climbing than that.

The most important feature of a climbing bike is how well it fits you. People tend to slack off on very flat roads, so efficiency losses due to poor fit don't have a big effect there. On a long climb, however, the difference between a good fit and a bad fit is very noticeable.

A stiff frame can be helpful on a long climb at higher speeds. If you use a granny gear, then stiffness is not so important.

As others have said, weight is not that important unless the climb is very long and the weight difference is several pounds or more.