I'll bet you an adult beverage that there will be little to no difference in hardness or pH (you have acidic water) but, more than 3-4 gpg of hardness, you have 7, will cost you more in hidden costs than a softener and some salt every month will cost you. The acid will also cause deterioration of metals in your water system and appliances while adding copper and any other metals in the system to your water; that can be a health concern, and especially for infants and young kids. And you can't maintain a water heater or other water using appliances against the problems they suffer because of hard or acidic water.
You said you had 1 ppm of iron, as little as .3 ppm can cause rust stains.
So post the new results.