When you register with the civil authority of the town you live in by empadronandote, you are merely registering in the town/city you live in. You do not need to show/proove any permission to actually be in the country. That is, you could be illegal even and still register - they allow this because it is beneficial for the town/city to have as accurate a head count as possible since the town receives benefits from the state based on these numbers (as far as I am aware from what I’ve been told). They do not cross-reference databases to make sure that you have legal permission to be in the country.
In fact it would not even make sense to do what I just wrote above, since the process to obtain the NIE requires first that you register with the civil authority and turn in the paper that confirms this. It is sufficient to use the passport as your method of identification. Later, if you get a NIE and then move and register in a new civil authority you can show either your NIE or your passport.
The process to register is pretty easy. Bring your passport and: either bring a person who is already registered (if you’re sharing a flat) with the authority at the domicile (making sure they have either their passport, NIE, or DNI with them); or you can make a photocopy of their form of ID and have them fill out and sign the application form if they can’t come with you - you just go and turn everything in. I did the latter with my Japanese roommate when he moved into a flat I was living in since at the time I was the only one registered at that flat. If you sign a contract for a flat, however, presenting the contract + your landlord’s DNI/ID + a utility bill under their name is sufficient and you don’t need to bother with having someone affirm you living there. For example, I currently am renting an apartment with my friends in the black and we can’t register ourselves to our flat because we have, for obvious reasons, no contract with the owner and no one here is registered to this flat. This time around I had a friend up the street help me register at his apartment so that when I renewed my NIE it would show an address in the current neighbourhood as opposed to the old one I was living in (I very well could have continued my NIE with my old apartment address, but for the purposes of the social medicine I would rather have access to the nearby center than have to go all the way to the one that pertained to where I used to live - another benefit of being registered IF you have the right to access the social medicine).
So you don’t need to bother with showing your belgian anything since the process does not presume that you are even (yet) legally residing in the country, merely there by some means.
Also, and importantly (I didn’t find this out until I had to register at my new address), you need to update every two years by re-registering.
Adios!