I'm sorry, I forget to welcome you, we're glad you joined.
Soft water doesn't mean it doesn't have chlorine, it has to have some additive which make it safe for human consumption, these additives are harmful to fish.
The ammonia won't evaporate with the water, although, I believe by adding water it would dilute it a bit The plant does not eat ammonia.
What eats the ammonia is a build up of beneficial bacteria. This is called cycling a tank. It's impossible to cycle a bowl of your size.
Here's a basic explanation of the nitrogen cycle:
First your ammonia (from fish waste and left over food) will rise.
In a few weeks you'll see the nitrite levels rise and the ammonia levels start to drop.
After a few more weeks you'll see the nitrate levels rise and the nitrite levels drop.
Ammonia and nitrites are toxic to your fish.
So until your cycle is complete, you need to keep the levels down with daily water changes
When you having readings of 0 on both ammonia and nitrites with some nitrates showing, your cycle is done.
I can't imagine your betta has much room to move around, please consider getting him a tank. You'll be rewarded with the most fabulous personality and colorful fins he can give you.
Edit: What's his name?