Some pilots swear by furniture polish like pledge to reduce the effect of the scratches.
Small, general aviation aircraft suffer from your same problem. When you are flying at altitude, with less atmosphere to protect you from the sun, small scratches and crazing can greatly intensify glare. It can also obscure other aircraft closing on you at between 100 and 600 relative miles per hour.
Check to see that the inside of your windshield is not covered in a thin, plastic film. Usually, windshields (as opposed to side door windows) are made from safety glass. They are thin layers of glass and plastic that have been laminated together. This strengthens the glass and holds it together if it were to break. Sometimes, the layer on the inside of the car is plastic. This inside plastic layer is very common when used in the following applications:
- As a UV blocking layer for sun protection.
- As a polarizing layer for the projection of Heads-Up Displays.
- As a conducive layer for electric defrosters.
- As a conductive layer for radio reception.
If the inside layer of your windshield is a plastic film, never use any chemical containing ammonia on it. That includes common glass cleaners like Windex. Also, as you have just learned, do not use an sharp or abrasive material on it.
Since the windshield is already scratched, get another RFID sticker to replace the one you removed. It can be an active or an inactive account number. The important part is that it will make the scratches less noticeable. Vaseline, mineral oil, or baby oil lightly applied will also do the same.
For extreme cases, a headlight lens cleaning kit may work on a small enough area.