If the stocking is just going to be-2 healthy adult Betta in a divided 10gal filtered tank-without live plants...
IMO/E-you can safely establish the nitrogen cycle with the 2 Betta-Provided that you make the needed water changes.
Since you have a test kit on hand-this will take the guess work out.
Once you acclimate them to the tank-monitor water prams daily and make a 50% water ONLY change based on test results of-Ammonia, nitrite 0.25ppm or greater. This water change is over and above your regular weekly 50% with vacuum that you will do for the life of the system.
But first-do base-line test for-Ammonia, nitrite, nitrate and pH on your source water-One with and One without your normal additives you will be using-like dechlorinator.
The base-line will help tell you if your source water has anything in it that you need to be aware of.
By testing with your dechlorinator-this will tell you if the product will cause any skewed test results that you need to account for-to avoid any unneeded water changes and stressing you out...lol....
Once you establish all your base-line numbers and aquarium-compare all these numbers-this will tell you if anything in the tank itself is causing any skewed results.
Plus by doing all these test will give you practice in using the test products.
Be sure and follow all the directions to the "T"...especially the time-lines between adding drops, number of drops and reading results to avoid user error.
If this is the API brand freshwater master test kit-Be sure and shake, bang on the table and shake some more on the #2 Nitrate reagent bottle-it tend to settle and can give false results.
It can take 3-6 weeks to establish the nitrogen cycle- You know your are cycled when test results read:
Betta can tolerate, adjust to most pH and it is best not to change the pH using chemical additives. It is safer to maintain a stable pH then having sudden extreme changes that can often occur with chemical additives due to the rebound.
As for the oxygen-your filter should provide enough surface agitation for gas exchange to support the beneficial bacteria for your nitrogen cycle.