Back on topic...
First, the issue here is not tap water vs. bottled water vs. RO water vs. deionized water vs. distilled water... it's all about mineral content.
Distilled water has the lowest mineral content, and will certainly not affect your developer activity or your processing. For this reason, there is no substitute for it, especially when making saturated solutions of chemicals for concentrates, etc. where any extra mineral content will affect the solubility of the chemicals.
That said, lots of tap water is perfectly fine for mixing working-strength developers; that is, if the mineral content is at acceptable levels or lower of the minerals that could negatively impact the developer; and, if the tap water is free enough of particulates. For this latter, we can filter, for reducing mineral content, we can often use an activated charcoal filter (Brita pitcher or the like) or boil the water. This is often all we need do to get water that is suitable for mixing our working developers.
The amount of calcium carbonate in good tap water can often be surprisingly high. Hard water will not only leave rings in your toilet, but often can increase the activity of a developer. That doesn't mean that it can't be used though. I have two different developing times for my darkroom in the U.S. (very soft, low-mineral tap water) and my "darkroom" in Vienna, Austria (wonderful mountain-spring water but rather hard); I need 5% more time in the U.S. to achieve the same density... as they say, "it's the water."
Iron and other contaminants (usually in well water) can wreak havoc with developers. If your water has these problems, then some other water source for your entire processing may be needed. Really, it's all about knowing what is in your tap water; then you can make an informed strategy.
Tap water often varies in quality throughout the year. If this affects developing, then standardizing on a bottled product (deionized or distilled) will eliminate the variable.
As for dissolved oxygen... I can't for the life of me see why on needs to worry about it. At room temperature, your aqueous developer solution will hold x-amount of dissolved oxygen. There's not really much you can do about that; distilled and deionized and all those other waters have the same amount of dissolved oxygen as well. The problem with developer oxidation is storage. As a developer scavenges oxygen from the water it is dissolved in, it gets replaced from the air. If we can prevent this latter from happening by airtight storage with very little trapped air in the container, then we slow down the oxidation of the developer, thereby extending its life. In a tray or a tank the dissolved oxygen level is going to be what it is. And, you can't mix it with water of any kind to get less dissolved oxygen in the solution unless your doing that in an oxygen-free environment.
One caution about softened water: Many water-softeners replace the calcium compounds in the water with salts. These are just as bad for leaving drying marks, etc. on you negs.
Finally, I really can't see why one needs to worry so much about the mineral content of their processing water leaving marks on their negatives. The real issue is how it may affect the processing. It's simple and more economical to use the distilled/deionized/RO water for the final rinse (with PhotoFlo in my case) and soak the negatives long enough to reduce the mineral content in the emulsion to low enough levels that drying marks will not happen. But, 30 seconds is not enough; a couple minutes or more minimum depending on your water's mineral content is needed. ...And, don't reuse the final rinse bath for subsequent batches! It now has dissolved minerals in it.
As for bottled water and the environment... don't get me started! But please, RECYCLE!
Best,
Doremus
P.S: Jaf, you seem to have found a good solution to your particular problem. Your cheaper bottled water should serve you well for all but making developer concentrates; you probably want distilled for that.