There is both ammonia and nitrite in the water so you are doing a fish-in cycle. It is better to do a fish-in cycle in a larger tank (there's more water to dilute the ammonia and nitrite) so I would move him over as soon as possible.
To do a fish-in cycle -
Test the water every day day and do a water changes whenever there is a reading for ammonia or nitrite above zero.
The water changes need to be large enough to get them down to zero.
Feed the fish only once every two or even three days. Less food = less ammonia.
Get Tetra Safe Start and add that to the tank.
If possible, get some live plants as they take up ammonia as fertiliser and they turn it into protein not nitrite. And bettas like plants. Even something like a few stems of anacharis left to float would help.
To do a fish-in cycle -
Test the water every day day and do a water changes whenever there is a reading for ammonia or nitrite above zero.
The water changes need to be large enough to get them down to zero.
Feed the fish only once every two or even three days. Less food = less ammonia.
Get Tetra Safe Start and add that to the tank.
If possible, get some live plants as they take up ammonia as fertiliser and they turn it into protein not nitrite. And bettas like plants. Even something like a few stems of anacharis left to float would help.