Aerating while injecting doesn't do any good at all since it just removes some of the carbon dioxide you are adding (putting it into the air as a greenhouse gas).
In theory, injecting carbon dioxide into the water will cause the pH to drop with no change in Total Alkalinity (TA). It is the exact opposite situation of what happens when carbon dioxide is outgassed where the pH rises with no change in TA. The TA does not change because when some of the carbon dioxide forms carbonic acid and then bicarbonate, this results in creating equal numbers of components that increase and decrease alkalinity so there is no net change.
CO2(g) --> CO2(aq)
dissolving gaseous carbon dioxide into water
CO2(aq) + H2O --> H2CO3
aqueous carbon dioxide + water --> carbonic acid
no alkalinity created yet (nothing was created to accept a hydrogen ion)
H2CO3 --> H+ + HCO3-
carbonic acid --> hydrogen ion + bicarbonate ion
hydrogen ion decreases TA by 1, bicarbonate ion increases TA by 1 (molecular unit, not ppm); -1 +1 = 0 for no net change in alkalinity
HCO3- --> H+ + CO32-
bicarbonate ion --> hydrogen ion + carbonate ion
removal of bicarbonate ion decreases TA by 1, hydrogen ion decreases TA by 1, carbonate ion increases TA by 2 since it can accept two hydrogen ion; -1 -1 +2 = 0 for no net change in alkalinity
The bicarbonate ion counts as one unit of alkalinity as it can accept one hydrogen ion, but the hydrogen ion counts as negative one unit of alkalinity since it is a hydrogen ion itself (technically, the concentration difference of [OH-] - [H+] is counted towards alkalinity since adding acid in the TA test essentially uses up this difference along with other chemical species that can accept a hydrogen ion down to the 4.5 pH of the TA test indicator).
The above analysis is also why pH and TA move together when adding a pure (strong) acid or base to the water. Adding hydrogen ion (i.e. an acid) lowers both the pH and TA while removing hydrogen ion (say, by adding hydroxyl ion from lye which combines with hydrogen ion to form water) raises both the pH and TA.
That's the theory, but in practice waterbear has reported seeing the TA increase over time when using CO2 injection so some other process must also be occurring at the same time. It could be as simple as the fill water replacing evaporated water since that happens in my own pool (without CO2 injection) where my fill (tap) water has a TA of 80 and my TA slowly rises over time [EDIT] or it can come from the "excess lye" in chlorinating liquid or bleach. [END-EDIT] I have a pool cover so outgassing is minimal and when the cover is open apparently the evaporation/refill is a greater factor than outgassing unless I do something intentional to make outgassing faster such as creating aeration. The rise in TA is slow in my pool -- rising from 80 to 110 over an entire season (around 7 months).
There are, of course, ways to increase TA as with adding baking soda (sodium bicarbonate) to the water as follows:
NaHCO3 --> Na+ + HCO3-
sodium bicarbonate --> sodium ion + bicarbonate ion
sodium ion doesn't change TA, bicarbonate ion increases TA by 1 unit; 0 +1 = 1 so alkalinity increases
The sodium ion does not count towards alkalinity, but the bicarbonate ion does. You can see the clear distinction between adding bicarbonate from a salt as above vs. adding it from carbon dioxide or carbonic acid which adds it from a weak acid which produces hydrogen ions that count against alkalinity resulting in no net change in TA.
To lower the TA, one must use the acid + aeration procedure (lower the pH, aerate and add acid to keep the pH low, then when the TA is where you want it stop adding acid and let the pH rise from continued aeration until the pH is where you want it), BUT the acid must be a pure (i.e. strong) acid, not a weak acid that doesn't lower the TA when added. So you can't use carbon dioxide, carbonic acid, nor phosphoric acid for this purpose. You must use a strong acid such as Muriatic Acid (hydrochloric acid, usually 31.45%), Sulfuric Acid or Sodium Bisulfate.
Richard