I think we can learn from history to answer this question. Not everyone is willing to fight. In cases where countries require all citizens to spend a number of years in the military, this is not always welcome. We must also consider the placement of Asgardia. How will the other nations view this?
With so many nations on Earth, the more advanced nations will certainly take note of Asgardia "crossing" their borders. We already see nations taking issue with simple satellites crossing borders or peeking in, for fear of potential espionage. Satellites have been lost this way. Now imagine their fear of a space station full of soldiers floating over their heads. This may also be a breach of the weaponization of space, as a military has weapons. With a full military in orbit, it would be simple enough to deploy into any location on Earth within hours via shuttles or drop pods. Earth-based nations know this, and would surely create countermeasures or plans for this. This turns Asgardia into a target then, with weapons poised on the station. In the station is hit, and entire nation could be wiped out.
Thus, I think it better if Asgardia remains neutral as is the current plan. Since citizens are permitted to have duel citizenship, if anything, the various nations of the world would be providing military training FOR Asgardia. This creates a broad mix of military and civilian talent with no cost to Asgardia.
I've stated in another thread the benefits of martial arts, and I think that it is the best way to go. Not everyone wants to fight, or is willing to learn, so they don't need to. But those who seek to defend themselves may take up martial arts. And as stated earlier in the thread, martial artists are respected, so having a space nation where much of the population knows martial arts should only increase the nation's standing in the public eye. It also generates a sense of pride for the citizenry. (Not to mention interesting martial arts tournaments between Earth and Asgardia, or just on Asgardia!)