No, it does not change the TDS, try and it and see 1st. See if you can soften the water and measure a KH difference.
CO2 can convert a TINY bit of HCO3 into carbonic acid, about 1 out of 400 molecules or 0.25%.
But this is insignificant, otherwise we could simply add CO2 and not need RO systems...........to soften tap water.
This is the part most often overlooked in these discussions which often are filled with a lot of misunderstanding and myth.
Plants can use bicarbonate if there's enough light and there's not much cO2, but generally this is an indirect method where they pump H+'s and acidify the HCO3 to CO2 at the cell surface. This is not the same as adding CO2 gas.
If you add say acetic acid or HCl to the water, this will react with the HCO3 and form water and CO2, but virtually none of the CO2 added will do this.