Wow I was totally under the impression that, within a short while, peroxide had broken-down fully and was no longer present.....you say they can stay suspended, but would they if the bucket had high-turbulence from a strong pump and there was some air-exchange for it to 'gas off' anything that becomes aerosoled/vapored(sorry, my chemistry chops obviously aren't not very strong!)
"4-6 days at room temp until most gasses are gone. What's left is acidic water high in oxygen"
WHOA! How big an effect on pH are we talking here? Not hard #'s but generally? I ask because I've been on the fence about what chemical pH-Down type product to get but they're all phosphoric acid based and I don't want to add phosphorous, so have been seeking a chemical to acidify my tap-water that's not phosphoric acid and this is the first one that seems it may fit the bill for me!! Do you think it'd be a practical approach for modest pH reduction? I have 8pH tap-water so even bringing it to 7pH is a HUGE increase for me, the change from 7pH to my ideal of 5.75pH isn't remotely as big as going from 8-->7pH! (as you say, tenths of a point are negligible, and there's massively diminished-returns when going downward from a poor 8pH tap!!)
(potassium lye sounds like it'd just be the same issue for me as phosphoric acid ie I'd be putting in a macro I don't want to put in....and you say "upping that pH kills a lot of algae", surely you meant lower it?)
If you happen to know any good 'ELI5'/beginner explanations on buffering I'd love to be guided there, the wiki is too-much for me to translate into IRL knowledge..buffering, buffering-capacity, etc, I go cross-eyed trying to understand it & translate it into practical application here ;P
Go with HCl as a pH-downer after your tapwater is left for a few days to evaporate most of the chlorine (place it in sunlight!). Otherwise citric or acetic acids (both buffering, so youll need a lot). Or nitric acid, but then tone down a little on the nitrogen when it comes to fertilizing.
pH effect of the peroxide is present, but I can't say how big it is. It depends on how much you use, how big your container is.. pH is maths, straight up. Add a little of this, or a little more of that.. then the equation changes, and so does the outcome. A few drops doesn't change a thing. A bucket does..
The lye was meant for the rainwater, which I read has a pH of around 4?! Then it should be upped. That kills the algae. Algae thrive at lower pH's and actively acidify solutions.
As for buffers: the simplest thing I can say is think about magnets. Everything is magnets. But everything that gets another magnet added to it, needs to take it from something else.
H2O (neutral) losing a magnet (H+) becomes OH- (see that it changes potential? Suddenly, there's a '-'). Water restores this balance: taking back as much magnets as it can, to get back to 0. So it takes back the H+. Water buffers itself.
NH4 can lose a H+, and it becomes NH3-. But it's in water! Water says: "no way hombre, I'm already as Neutral as Switzerland, take your own magnets back!"
NH3- says: "Man, I'm kind of negative now, would you mind if I take back my magnet?"
Water says: "Sure, here ya go!"
NH3- becomes NH4 again.
But then the water splits again, remember, it buffers itself, so in the fragment of a second, there's OH- and H+. The OH- pulls the magnet away from the NH4.
NH4 loses it's magnet, it becomes NH3-.
OH- takes up that magnet, becomes H2O again. But suddenly, the H+ that has left the water, has no place to go. Luckily there's NH3- willing to take it in.
This happens a few million times a minute in a ammonia solution.
This happens with all 'weak acids' or buffers.
Remember that water is weird. It's one of the hardest to explain parts of chemistry. Simply put: it does things it shouldn't do, but we're all thankful that it does.
With strong acids, like HCl, there's not that much of attraction going on. Cl- just says: "I never liked that guy H+, water, please keep him away from me".
And so HCl becomes a strong acid. In a solution, H+ and Cl- will never bond again. Hence there is no buffer. They break apart. They filled in the papers, water can keep the kid.
We talk about "H2O with acid" but when a surplus of H+ occurs, H2O becomes H3O+ and starts buffering itself with itself. H2O+H3O <-> H3O+H2O. (notice that one H has skipped places).
Khanacademy.org is the best place to learn about chemistry (and biology) for free. If you can withstand the dorky presenter.
That's the best I could do in this limited time. If you'd like to know more, send me a PM. That's easier for me to read and answer during my daily 4 hour commute.
) so I can go over it a bit more, thanks a lot for that 
