SE retains the last file type open so it should stay at Solidworks Part Document until you select something else. Of course you still have to select an SE template after opening a non-native SE type.
You can assign the default "open with" to the "Edge.exe" to open automatically in SE.....works for me.
[alternatively...assign the "SW2SE.exe", if you don't want to have confusingly similar icons to other SE assignments.....like DXF/DWG, etc....]
Design Manager Streetscape Limited
Solid Edge ST10 [MP0] Classic [x2 seats]