You can transfer some of your established filter media, no more than 25% max of sponge media and a small amount of ceramic noodles/bio balls to the betta tank filter, or if it’s a sponge filter tie the sponge media from established media around the new sponge media with thread or elastic band, this will pretty much instantly cycle the betta tank.
Even just small handful of substrate from established tank to the betta tank will help as well.
One or both of these methods will have some of the bb you need and will spread all around the tank and filter quickly.
I completely forgot you had other tanks