I wouldn't have a 10 gal without cycling it. I have two 10 gals each
with one betta and a 6 gal with one betta and all the tanks are cycled.
They benefit greatly from a cycled tank. You don't HAVE to cycle it but
you will HAVE to do 100% water changes instead and that's just a ton
more work for you. Plus, you'll be testing your water far more often
and have to be regulating all the water parameters at each 100% water
change. (It's a lot easier to keep the pH and temp stable when you're
only changing 20% of the water instead of 100%.

Bettas should not be exposed to any ammonia or nitrites. If anyone
tells you differently, they haven't battled the fin rot. Even exposure
to .25 ppm over time can cause a great deal of damage in these fish. .

I'd get filtration and cycle the tank (without the bettas) without a
doubt.