I can't keep regular neon tetras alive either.
Black neons seem hardier, ditto rummynose, lemon/pristella, and some of the other larger tetras. You have to watch serpae tetras, as they're nippy, and some of the ones like BA tetras or similar just need a lot more room, like a 4' tank.
Suggestions might include trying a bigger tank - neons and smaller tetras get skittish if there's not enough swimming area; more cover - they like fake or real plants; and, trying a different source that might have healthier stock or have kept them for several weeks successfully as they have a high dieoff in shipping.
- On the bigger tank, I see you're trying in the 29. That should be enough room - but I'd still consider adding more plants for cover. Based on some videos I've seen, in the wild they live in semi-murky shallow streams/water with lots of grass and weed type plants.
The so called 'neon tetra' disease wipes out a lot of tetras and IIRC there's no worthwhile meds to treat it - in my opinion, spending $15+ to treat $2 fish doesn't seem worth it; others might disagree.
I know you say the betta is not killing them, but are you sure there's no marks on the dead tetras? If not, then probably it's disease. If there are marks, the beta may be bumping them off in the night when you're not watching. They can be sneaky about killing sometimes, even though they seem calm when you're watching.