Are you talking about a server that is completed coded by the person running the website, and as a result prevents anything but his applet from connecting?
or, are you talking about a widely avaliable package such as Unreal or Bahamut running on it..
If you are talking about the first option, then it would be highly unlikely this could be encorporated since mIRC would have to interface with some sort of applet which could easily prevent it doing so anyway.
And the second option, then more than likely the network is sending a VERSION/FINGER request to identify your client software or the server is setup to allow only a specific port to be connected to, not default (6667-6669,7000) or, the server is checking for a specific gcos,identd - in which case all the above can be "changed" to make it seem like mIRC is connecting from the applet..
HOWEVER, You more than likely would be able to find help in evading this "ban" they have set since it is their wishes regardless if its a good one or not.