Hydrogen peroxide is universally biocidal, meaning that enough of it will kill any organism. It’s a reactive oxygen species so it reacts with most organic compounds, altering them, and therefore reducing the compound’s function.

Hydrogen peroxide is produced by immune cells in the human body to kill microorganisms and they use catalase also to reduce damage to the human cell making the hydrogen peroxide. The way enzymes work is that they can process a certain number of molecules per time. This means, based on the number of enzyme molecules present in the bacteria, there is a maximum amount of hydrogen peroxide that they can degrade in a given amount of time. However, if they are flooded with hydrogen peroxide they won’t be to break enough of it down in time and the cells will die. In summary, if enough hydrogen peroxide is added to the bacteria, they will certainly die.

Enzymes are catalysts: they allow chemical reactions to occur more swiftly and requiring less external energy to happen, but they can't cause a reaction that doesn't happen normally and don't get used up. Enzymes do eventually degrade, but this degradation is not related to how often they are used.

Catalase detoxifies hydrogen peroxide by turning it into less toxic compounds of water and elemental oxygen. Living things protect themselves from hydrogen peroxide using this enzyme. Contact with hydrogen peroxide will cause damage to anything it can react with, but the enzyme will slow down the process by neutralizing the poison. Bacteria that can make catalase therefore are greatly more tolerant of hydrogen peroxide than they would be without the enzyme.

If hydrogen peroxide will not work, I might suggest using a different pH (although be careful not to use something that will corrode your water well!). Bacteria have a range of pH that they will tolerate, and exposing them to a pH outside of that range will kill them (different bacteria tolerate different ranges, of course). You might also try iodine.