Betta + black neon tetras?
[First off, YES. Bettas have been kept fine within community tanks! I just don't know about this specific container!]
I don't know what is up with the betta tank, but it is completely infested with algae and keep turning green within days of cleaning. I've given up on that thing. Malice needs something exotic. (I have another tank of the same fashion for my ACF's feeder guppies, and their tank never even gets a green tint! Let alone so cloudy I can't see my fish!)
I have a 10 gallon reservoir with much better filtration and no algae problems in the same room. In the cistern are only three black neon tetras and a black snail. Would they fight? There are tons of places for the tetras (or Malice) to hide. Is nearby any way to introduce Malice in with the tetras to be paid the transition easier like with other animals?
As you have found out - your tetras and Betta should be OK together in that size tank.
Your problem beside the other tank going green? Too much light. Move it away from the windowpane or shade it somehow. That will cut down on the algae growth. The green wont harm your fish, but it does look ugly.
So make sure you don't overfeed as in good health as cutting down on the lighting.
Bettas, obviously, are thoroughly territorial, and agressive fish. In the wild, they tend to choose their own part of a pond (cave, crevice, a bunch of plants, etc.) to be their "territory". When you put a betta into a tiny glass bowl, they markedly can't get out, but they still consider it their own private area. When I say private, I imply they do not want any other fish trying to get into that area, or taking that area.
When a betta is introduced into a container, believe it or not, they will find an area to call their own, even if its merely a patch of gravel in your ten gallon tank. Mainly, they will try to defend their nouns from other territorial fish, such as other bettas. Although they may consider the schooling tetras as a pest, they generally won't bother them, because schooling fish tend to stay towards the surface of the water, and they do not claim specific territory. Thus, the betta does not consider them a threat or foe.
The betta may nip at their fins every once and a while, but this is more of a notice bite, nothing compared to a full fight of betta vs. betta! Also, the chance of your betta getting impossible to tell apart fish twice on a regular basis is quite slim.
Simply put, your betta will soon realize that the Tetras are not a threat to him, so he WILL leave them alone.
I hope this help!