Good point. Since you know the weight of the new axle (it's probably stamped on the box) - can't you add about the same weight into the trunk? A few bags of sand or weights or one mother in law should do fine. People sell weights for cheap on craiglist all the time.Apparently there's a science to lifting the tiny car with such weight bearing work to be done. You will be shifting a good amount of center of mass when you remove the axle, so have to choose the appropriate support points. I have yet to figure out where in the driveway to set up. I'm really spoiled with regards to driveway space but I've taken up a lot of it over the years. I have a spot in mind where it won't block much but I'm a little worried I might be cramped there.
Putting a jack under the cross-member probably would not hurt either.