Alkalinity is the amount of acid required to reduce the pH of something (in this case a liter of your water) from the pH at which it exits the tap to pH 4.3, 4.4 or 4.5 depending on who is measuring alkalinity. You are interested in it because you will need to move the pH of that water to a pH near 5.5 in the mash. If the labs were to measure pH to 5.5 instead of 4.3 - 4.5 the alkalinity number would be more useful to you. So given that you don't have lab service why not simply measure it yourself? This is, in essence, what you proposed. Take a liter of your water and add carefully measured amounts of lactic acid to it until you reach mash pH. You can then, if you wish, note that 88% lactic acid is 11.5 N at pH 5.5 meaning that each cc delivers 11.5 mEq of acid (you will want to dilute the lactic acid if testing 1 L but you could work with the full volume of water - it's cheap). The alkalinity to mash pH is the number of mL used to reach mash pH times 11.5 times 50. The alkalinity as usually given on a water report is about 1.1 times this. You can also enter the amount of acid you used into a spreadsheet with the volume of water you tested and then fiddle with the alkalinity entry (unfortunately most of them will have you fiddle with the bicarbonate entry as they don't accept alkalinity directly) until the pH matches the pH you reached. Converting the bicarbonate number to alkalinity then tells you the alkalinity of the water. This will be a rough number as you don't really know the strength of the lactic acid (not a problem as we will see in a minute), using bicarbonate as a proxy for alkalinity can induce errors if the pH is above about 8 and you will have made measurement errors.
If you had a laboratory's alkalinity number you would use it to compute the amount of acid you would have to add converting the result to mL 88% acid required. But you already know this. What I would suggest is drawing the volume of water you are going to use and adding lactic (or phosphoric) acid to it until the desired mash pH is reached. If you undershoot (too much acid - too low pH) simply add more water to get back to where you want to be. You don't even have to measure precisely. Your goal is to hit mash pH. You now have before you a volume of water with 0 alkalinity to mash pH. Enter 0 bicarbonate in your spreadsheet program and proceed with the grain additions. If extra acid is needed for the grains or if the grains supply excess acid the program will tell you that you need, respectively, more acid or some alkali. If you used 20 mEq of acid to treat your brewing water and the program tells you that you need 5 mEq of base to compensate for dark malts then throw out the water and make another batch using only 15 mEq of acid this time.
It would be a good idea to keep track of the amount of acid used in order to estimate your alkalinity so you will have some idea as to what it actually is. If it is high you may wish to consider methods of decarbonation other than adding acid.
