That's not possible because you are not stateless. You could presumably easily return to your country and serve in the military if you wanted to, that's pretty much the opposite of statelessness.
You keep saying that you cannot renew your passport but that's not quite the problem here, at least from the perspective of the authorities. The fact is that you're trying to avoid conscription and governments do not necessarily want to make that easy, even for foreign citizens.
In the EU, being called for military service can be grounds for subsidiary protection but only if you are fleeing military draft in a country that is at war, you can show that you would be forced to commit war crimes or face unreasonably harsh punishment. It's not the case if you merely find military service inconvenient.
Also note that Switzerland (which is not in the EU) is a country with a strong tradition of mandatory service in the military. All Swiss men have to serve for several months of mandatory training when they turn 18, get a service weapon, and then go back every year or every other year for a few days until they turn 40 or so. If you get an exemption (even on medical grounds), you have to pay a special 3% income tax, which Swiss people frequently describe as a way to contribute to the defense of the country in another way.
Military service is therefore much more important there than in neighbouring countries like France or Germany. Culturally, I would not expect much understanding of your wish to avoid service from Swiss citizens.
