The nitrogen cycle will happen on its own as long as the beneficial bacteria needs are met......That being-oxygen, food source and surface area....
You can safely establish the nitrogen cycle with a healthy Betta in the filtered 5gal tank provided that you make the needed water changes...with or without testing products on hand....
Regardless of what you keep the Betta in...It will produce byproducts and will need water changes-the only difference will be the bacteria you are colonizing to establish the nitrogen cycle...
To safely cycle with the Betta in the filtered 5gal-Make weekly 50% water changes with substrate vacuum or stir and dip method-this will be the water change schedule for the life of the system to maintain water quality once the nitrogen cycle has established.
For the next 4-6 weeks-without testing products on hand-Make a second 50% water only each week.
If you have testing products on hand-Base the second 50% water only on test results of-Ammonia, nitrite 0.25ppm or greater.
Once you start reading nitrate 5-10ppm without any ammonia, nitrite spikes for several days-you are most likely cycled-this can take 4-6 weeks.
Remember that some additive used-especially dechlorinators and ammonia removers can cause false ammonia reading since they change ammonia to ammonium and the test products can't tell the difference-so not to make unneeded water changes stressing the Betta....
Adding lots of active growing stem and floating plants can also change the cycling process. As well as a low pH....