Perhaps by understanding what is actually going on in the water might help. Total Alkalinity is the measurable carbonates and bicarbonates in the water that are part of the buffering system. (For simplicity I am going to ignore cyanurates from stabilizer and borates from sodium tetraborate 'water enhancers' that might be present also). The other part of this buffer is carbonic acid (essentially carbon dioxide dissolved in water--same a seltzer). When you lower pH you lower TA because you convert some of the measurable carbonate and bicarbonate into carbonic acid. It doesn't matter HOW you add the acid, just that you add it and lower the pH. Now the trick is to get the pH back up without causing the TA to rise also. If you add a chemical that is alkaline you will raise pH but also convert some of the carbonic acid back into carbonate and bicarbonate and therefore your TA will rise also. Soda ash (sodium carbonate) will add a lot of carbonate to the water and will actually cause the TA to rise faster than adding sodium bicarbonate and it is possible to end up with a higher TA than initially. If you add sodium hydroxide (lye) or sodium tetraborate (borax) then you will cause less of a rise in TA as the pH rises compared to soda ash.
The problem becomes how to remove the excess carbonic acid and cause the pH to rise and the answer is simple. Aerate the water. This will cause the carbon dioxide to outgas and the pH will rise. (Just like shaking a bottle of seltzer to make it go flat.) Aeration is easy to accomplish by using a fountain or other water feature, turning the eyeballs up to break the water surface, or using an air compressor to cause air to bubble up through the water (much like an airpump and airstone are used in an aquarium). Even a rig made from PVC with a 90 deg elbow that causes the water to shoot into the air and then return to the pool placed in a return can be very effective.
The biggest problem with the process is that it takes time to gas off the carbon dioxide, depending on how vigorous the aeration is. The 'slug method' does not work and only creates pockets of very low pH that can damage pool surfaces. To effectively lower TA quickly and keep it down drop the pH to 7.0 and test TA, begin aeration and monitor pH. When pH starts to climb add more acid to drop pH back to 7.0 and test TA. Repeat until TA is at target value and stop adding acid and just aerate until pH rises to target value.