Personally, I wouldn't cycle something smaller than 5 gallons. That small volume of water is very susceptible to rapid changes in water perimeters making it extremely difficult to keep a cycle. I would recommend just doing a 100% change every week. Without a cycle, a filter is basically useless as they don't really clean the water, just facilitate bacteria growth and aerate the water (which doesn't matter as bettas breathe from the surface). Also, that jump start "bacteria in a bottle" is useless. It won't harm the fish, but there is no actual live bacteria in there like they say. There is no source of food, air or light inside a bottle, so any good bacteria in there will be dead long before you add it to your tank.
And you can still use the cichlid tank to start it. The bacteria isn't in the water. You can use some gravel or filter media. Any minute amounts of salt transferred will be fine. And as long as what you're using isn't disgusting looking, I'd feel fine using it to cycle my tank.