Mmmm... Well maybe the Walmart chlorinating liquid had a lot of excess lye in it, but it would have to be an awful lot to rise as much as you are seeing. I use Hasa in my pool and it does contribute to a rise but not by that much -- adding acid every week or two to move down 0.2 pH units (7.8 to 7.6 usually). With your TA down to 60 ppm after the fill, that would very much lower any carbon dioxide outgassing so I wonder what your source of rising pH is. Since you have a rising CH issue from evaporation and refill, if you do replace your water to dilute it over time then you could use pucks sparingly and get some pH lowering from that, but as you point out it will raise the CYA if you don't have water dilution.
The borates would slow down the rate of pH rise, but wouldn't change the total amount of acid that would need to be added. It's a pH buffer, but it wouldn't change total acid amounts unless you had low-level algae growth that it helped control so that it lowered chlorine consumption. What's your daily FC consumption like? Is it around 2 ppm FC per day?
You had mentioned your neighbors and co-workers before, but I thought they were using Trichlor pucks/tabs in which case they'd have the opposite problem and might need to use pH Up to keep the pH (and TA) from dropping too much. Do you know the pH and TA of your fill water (you can always test it)?