Well, your cycle has already started.
What kind of tetra is it? You don't really have to worry about the betta, and just 1 tetra shouldn't have much of an oxygen demand, how big is the aquarium?
I say kill the current, and if you see the tetra gasping, then turn the current up a bit and see what happens, keep turning it up until the tetra stops gasping. If you think that the current you stop at is too strong for the betta, consider floating something (a leaf etc, anything flat really), that he will be able to build his bubble nest under.
Goodluck resolving this lol
Gasping is when the fish is swimming at the surface of the water with his mouth actually touching the surface, looking as though its trying to breathe air from the atmosphere (which is usually the case if there isn't any oxygen in the water)
As for the cover, yes it would, but make sure you have your clip (the thing that keeps it upright) as sometimes they slightly overflow if they hang backward.
It is possible for you to buy a small 2-5g aquarium for the betta? it wouldn't even need a filter, just maybe a silk plant and a little terracotta cave
Where do these folks get their information, and how do they not kill all their stock?