I came across the info below, which strongly suggests that greystoke was right and aeration should remove all the bicarbonate from solution in time, and addition of acid finish off any residual bicarbonate ions.
Certainly, I've only been aerating my tap water for around four hours now, and the pH has already increased from an initial 7.98 to 8.39:
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H2O + CO2 « » H+ + HCO3-
Carbon dioxide, hydrogen sulfide, and ammonia are soluble in water under certain conditions to the extent of 1,700, 3,900, and 531,000 ppm, respectively. Rarely are these concentrations encountered except in certain process condensates. In a normal atmosphere, the partial pressure of each of these gases is practically zero. Consequently, the establishment of a state of equilibrium between water and air by means of aeration results in saturation of the water with nitrogen and oxygen and nearly complete removal of other gases.
As the equations above show, ionization of the gases in water is a reversible reaction. The common ion effect may be used to obtain almost complete removal of these gases by aeration. If the concentration of one of the ions on the right side of the equation is increased, the reaction is driven to the left, forming the gas. In the case of carbon dioxide and hydrogen sulfide, hydrogen ion concentration may be increased by the addition of an acid. Bicarbonate and carbonate ions in the water will form carbon dioxide, which can be removed by aeration.
In a similar manner, an increase in hydroxyl ion concentration through the addition of caustic soda aids in the removal of ammonia.
Gas removal by aeration is achieved as the level of gas in the water approaches equilibrium with the level of the gas in the surrounding atmosphere. The process is improved by an increase in temperature, aeration time, the volume of air in contact with the water, and the surface area of water exposed to the air. As previously indicated, pH is an important consideration. The efficiency of aeration is greater where the concentration of the gas to be removed is high in the water and low in the atmosphere.
Jack you should try to ferment it to achieve stability rather than bubbling IMHO.