Conclusion first, the grammatical topic of the second sentence is not 問題(は). I know this is unusual, but in this case, adding 問題は at the beginning of the sentence will dramatically change the meaning of the original sentence!
There was at most only a week's worth of drinking water. / There was drinking water that might or might not last for a week. / There was some drinking water, but I was not sure it would last even for a week.
The problem was whether or not there was a week's worth of drinking water.
I think you already know how the second interpretation works. ～かどうか forms an embedded question, "whether or not ～". I was initially unaware of this, but the second interpretation would also be perfectly valid in a certain context. For example, if he knows he will be rescued after a week, and if he has everything to survive other than drinking water, then the only problem left is whether or not there is enough water for the next seven days.
But the actual story is not like that, and that's why you asked this question, right? :) Perhaps he had no idea when he would be rescued; he might have to wait much longer than a week. Thus the correct interpretation is 1, and you must not add 問題は at the beginning for this sentence to work. You have to be able to interpret this sentence as-is, without adding 問題は.
So, how does the first sentence work?
～かどうかだ or ～かどうかです has a usage that is a bit different from "whether or not ～". This is a little colloquial and I don't know how to normally translate this, but the basic idea is something like "may or may not", "fifty-fifty chance", "not certain yet", "depends on chance", "on the borderline between success and failure", or such.
From here, you can see Mt. Fuji if you are lucky. / Sometimes we can barely see Mt. Fuji from here. / There is a chance we can see Mt. Fuji from here.
He is on the borderline between pass and fail.
I'm in a hurry, but I may or may not be able to arrive on time.
Sorry for the loose translation, but I hope you now understand how to interpret the original sentence without adding an imaginary topic.
Finally, if we do need to start this sentence with 問題は without largely changing the original meaning, we can say this:
The problem was the fact that I had only a week's worth of drinking water (at most).
I simply enclosed the "かどうか" part with a nominalizer(?) だということ.
(EDIT history: Rewrote almost everything, but my opinion is the same. I hope I have explained this issue better now.)