I actually have never been asked for a residence card in any of my Internet subscriptions and I am currently in the process of changing my Internet connection to a 5th provider (edpnet) (Belgian ISPs are unfortunately not yet on the same level as those in the neighbouring countries :( ) . So I am a bit surprised. Maybe you should not mention your status. I assume you have a Belgian bank account. As said, I am currently considering edpnet, but until now I had Dommel, Scarlet (The 2 others no longer exist).
If they really make it hard for you to get connected, I could recommend Mobile vikings. In my opinion they are the only Belgian ISP company that already have entered the 21st century. I would be surprised if they wouldn't be able to help you. It is prepaid internet. It is going to be a bit expensive though. You would need to buy packs of 2 GB for 12 Eur.
Another trick you could use is to resort to FON. Belgacom provides FON points on all their ADSL routers. So if you buy a FON router and install it on an internet connection abroad and you immediately have access to all Belgacom ADSL routers.
But again I think that you should simply just not mention your legal status and you will be okay to get a subscription at any Belgian ISP. Maybe you should not go to any of their stores, but try to apply online. The shops you see everywhere are really useless.