It seems to me that the "mushroom" effect would occur no matter what corker you use, as long as it's a Belgian cork going into a Belgian bottle. The top portion of the cork will just naturally be wider than the neck of the bottle.
One thing that I have learned to work on for each batch, is to adjust the depth of the plunger to leave as much cork exposed as possible while still being able to pull the cage down over the rim of the bottle, then tightening the cage.
There are no good tools readily available to pull the cage down and tighten, and also leave you with a large enough wire loop to easily unwind the cage when you want to open it. The ones with a wooden handle that are available at LHBS work great, but they leave you with a loop that is very thin and it hurts to open it. My go-to tool for this purpose is a 1" hole boring drill bit with a long stem that is about 1/4" in diameter. It has plenty of length to grip and pull down on the cage, twist it 5-6 times to tighten, and leaves a nice round loop in the wire which is easy to open.