In my experience, cycling with fishfood or shrimp does not take any longer than with pure ammonia. It's just a smelly, nasty mess which can also promote a Saprolegnia mold infestation.
When cycling, what takes the longest is waiting for the nitrifying bacteria to fall out of the sky into your tank. Whatever you use to provide ammonia takes about the same amount of time if you're working in the range of 2.0 to 4.0 ppm, as is usually advised..
Using the right bacteria in a bottle
can cut your cycling time in half, if you're lucky. But not much more than that. Nor is there any advantage to quickening your cycle that I can see.
Once you cycle a tank/filter, you'll always have live bacteria to seed further tanks. You won't need ammonia after that unless you plan to grow a large colony for heavy-stocking purposes, more than a typical Betta environment.
Sorry this got onto the AB thread, but it might be useful.