I'm using a tetra also on my 10G, so far its OK, but I'm paying attention to it b/c its doesn't move much water. I have RCS(shrimp) so I dont want them to get sucked up.
Now for a betta, they dont like lots of flow. Some people have planted tanks w/ no filter at all and no issues, but this is against common opinion.
Is it necessary, no, if you had other fish then I would say go for the AC
, w/ a betta he may not like an AC
filter, even though the flow is adjustable.