If you use the procedure to lower the TA substantially, then your rate of pH rise will drop a lot as well. Read the HowTo on Lowering Your Alkalinity. There is no avoiding the amount of acid you need to add to lower the TA. You can use the "Effect of adding chemicals" section near the bottom of The Pool Calculator to calculate how much acid it will ultimately take to lower the TA, though this isn't added all at once (i.e. follow the HowTo procedure).

It takes around 25-1/2 fluid ounces of full-strength Muriatic Acid (31.45% Hydrochloric acid) to lower the TA by 10 ppm in 10,000 gallons.