The weedy portion seems like it will go a bit quicker than the sod as a) there aren't as many plant roots to deal with and b) what there are for roots don't seem to extend as deep. The downside is that this portion of the ground nearest the house is also the rockiest, so maybe it'll be a wash.
For reference, the area I'm planning on digging up is about 600 square feet (60 x 10).