Using a TDS meter to measure water quality
TDS meters measure the conductivity of the water, which can help identify salts or other impurities which have positive charges. However, common disinfection by-products in water, such as chlorine, iron, chloramine etc, do not impact the conductivity of the water. So they would not be picked up by a TDS meter.
The best way to test the effectiveness of the shower filter would be to use a chlorine colorimeter or chlorine test strips.
Aside from testing water, the other things to look for would be: No chlorine smell in the water, less dry itchy skin, stronger nails and even soap which lathers up more, to create more bubbles.