Congratulations on starting! That's most excellent.
One of the very difficult things to learn when starting out is the difference between "pain" and "soreness". I assume you're using the word "pain" because it hurts - and that you're not experiencing those feelings that come along like "Wow - I haven't used that muscle in a while!". If that's the case I would not try to do workouts ".....with or without pain." Pain shows up when we need to change something. And if something doesn't change it almost always gets worse. :-( Sometimes just backing off to let the body catch it's breath is all that is needed.
In my opinion I think the real goal is to develop a long term exercise habit. I think repeating W1D1 makes sense. See if you can do that and if the Achilles is happier. How long is C25K? 9 weeks? So what if it takes 10, 12 or 20 weeks? There are a MILLION 5 Ks out there. You might make it to and through your Oct 5K and that would be awesome. But if you continue to have troubles and you push through the pain, you still might get through the Oct 5K, but you may be miserable, dislike running, and even if you want to continue it might take many months to truly recover from pushing too hard.
So I'd say focus on establishing a long term exercise habit. If it takes extending the time for C25K so be it. And in that spirit I think doing it in the pool is a fantastic idea. It could relieve the stress on the Achilles enough to help it heal if in reality it truly is injured. Be aware, though, that doing it in the pool is not the same as doing it outside. In the pool you're not bearing your full weight and outside you are. A key objective of the C25K is to get your body used to the stresses and efforts of running 5K - bearing your full body weight.
Congratulations and keep it up! Dealing with niggles and having to adjust plans, unfortunately, is a major drawback of this totally addicting hobby. :-)
"Kick off your high heel sneakers, it's party time."