I suggest the following:

Keep the first rule:

1. If both players pass consecutively, then they cannot pass again until each player has placed one more stone on the board.

And add this rule:

2. If a player has no legal move or if his only legal move is to place a stone inside his own uncontested territory (that is, in territory surrounded by his own stones and which does not contain any stones of the opposite color), then the game is ended and whoever has more points is declared the winner. All stones on the board are counted as alive (this isn't right, but it will make the programming easier for Fencer ;-)

These rules would force an end to the game. I don't think they would lead to a change in strategy. And players would not be forced to fill in their own eyes. What do you think?