Pinus mugo is good at compartmentalization. This means that one branch is connected to one side of the root system as opposed to all branches connected to the trunk, which is connected to all the roots.
If you lop off some branches, the roots they're connected with might die. If you lop off some roots, the branches they're connected with might die.
I think(!) that dying branches on a mugo don't necessarily draw energy from the rest of the tree due to that compartment system. I would try to keep as much foliage as possible, because you can't really be sure what's connected to what. You might just snip off the only viable branch left, without knowing it. I personally would leave it as it is, and just see whatever makes it.
Someone else might have a different opinion though. It might be wise to wait for others to chime in.