No problem injecting CO2 into basically zero KH water. I do it every day.
With very low or no KH, tank KH/pH swing during the week is minimal. In fact, I still use my pH controller as my KH/pH remains stable. Right now I am dropping pH down to 4.90 during the lighting period. Degassed is about 6.3 pH.
If KH was higher, soil would deplete that during the week and it would be more difficult to control CO2 injection. As KH goes down, so would pH. So relative pH drop would be difficult to measure properly. If that were the case, better to set a constant CO2 flow rate and not worry so much about where pH is. And yes, that is more difficult to control.
As to GH boosters like Seachem Equilibrium mentioned above, you are much better off just getting some CaSO4 and MgSO4. All GH boosters are basically a mix of Ca, Mg, and K. You don't necessarily want or need the extra K. Too much can interfere with uptake of other nutrients. You can also fine tune the mix of Ca to Mg. I've had more success with higher Mg than most boosters provide.
As to pH crashes due to low KH/pH, pretty much a myth. Here is a great article about it...............
As to Brighty K, it's basically KHCO3 mixed with water. If you knew how much KHCO3 you can buy for the cost of one bottle of Brighty K, you would never buy it again. You are pretty much paying for water in a bottle.
Hope that helps, and good luck with the tank.