Does this law apply to suing the government? 

If you are suing the government because they committed a wrong, police shooting, tore down your house, highway truck slammed through your living room, etc, sure. But if you are suing to overturn a law you don't like, the loser pay rule shouldn't apply. 

I didn't live in Alaska in 2003, so I am not familiar enough with the law to answer that question.