There seems to be a rather serious bug with the Austrian missions. If you get the Austrian special PU over Bohemia via the Improve Relations option (haven't tested it for enforcing through war), after Friedrich III dies, Bohemia just becomes independent because of a faulty event. I've had this happen in two separate games and tested it by reloading a save at least 20 times.

In my case, they have +200 opinion and 0% liberty desire. They seem to get the "Election of a New Ruler" event because even though they become a PU they retain their Bohemian Elective Monarchy special Tier 1 government reform and always pick "Choose one of the House of Habsburg", however that gives them a brand new monarch of the Habsburg dynasty, thereby severing the PU.

The workaround for anyone also annoyed by this is to go into the console in your latest autosave before Friedrich dies and do

tag BOH
kill HAB
//select the option that retains the new Austrian monarch as king//
tag HAB