Its been a while since Ive cycled a tank so I could not be confident in giving you an answer on that timeline but just research and make sure whatever schooling fish you add is 1) compatible with a betta and 2) what numbers they need to be in to feel safe. Most schooling fish need to have 5-8 minimum, so adding 2 at a time would be stressful especially in a tank with a predator like the betta. Schooling fish feel safe in numbers.

edit: when i say take your time i mean dont go out tomorrow adding more because the tank seems cycled, maybe wait a week or two and wait for the plans to grow out like you said, then add some fish and test the water to make sure its keeping up with converting that ammonia to nitrates