Carnie: you have some contradicting asumptions here...

You talk about the end of brainking - there is no defined end, only in the mind of some fatalists. But even if brainking 'ends', then you would not 'time-out', and even if you would, it wouldn't matter anymore. So where is the problem?

Also, there is a resign button at the bottom. Keeps you from timing out, if it is so important for you.

Did you contact your opponent and ask for a draw (or offer a draw in the game)??

Finally, which part of 'Time per move: 7 days' did you not understand when you signed up for the game? It is well known that many players fully use the alloted time; and it has been discussed about a gazillion times in detail here. If you don't want to play that slow, don't sign up for such games.