And thus, the JSDF fought here ...
Well , there is a neat and reality based way of circumventing the no military clause: just send a UN force for peacekeeping proposes. With Japan as the only UN country in that world, it shouldn't be hard to pass that resolution
@Waifhunter
The GATE situation is somewhat diferent. Technically, as the gate opened in Japan on there, the land on the other side of it is japanese territory as far as international law is concerned ( as no land, sea or air borders were crossed and as it wasn't declared international territory ) and the JSDF can do whatever it wants because of that.
In here ... Well, Japan has recognized the nations in the other world as political entities to the point of making commercial treaties. So, either the bar to external deployment is waived ... or a humanitarian / defense of Japanese citizens "reason" will appear.