Easier to do a cycle without the fish but not recommended by many fish experts I've talked to. The enzymes and bacteria in the betta's digestive system (its poop) are a neccesary part of the cycling process for it to complete properly.
You always want to use the type of fish you are going to house in a tank during the cycling process. I have had many tanks in my lifetime (new to bettas tho) and have always put fish in them during the cycling process to make sure it completes properly. Just make sure your test levels are safe to add the fish. It is ok if nitrate levels are slightly above normal during this time.
It may stress the fish a little at first, but it will help your tank cycle properly and lead to happier fish in the long run.
If bettas are different for some reason, someone please let me know. But, I just finished cycling a small tank and had a betta in it about 2 days after I put it together. Cycling finished in about a week for a 1 gallon desk tank.
I am concerned about the no-filter, no-heating in a 10 gallon tank tho... you may end up with dead fish.