We have two dogs: an 11 yr old boxer (super arthritic and slow, but she loves to chase goats even though she is so slow now that she can't really catch them or do them any harm) and a 2 yr old South African Boerboel. The boerboel was/is our problem with the chickens. When we first got them, he was still a puppy and we figured out pretty quick that he has a high prey drive. This breed is very large (he has still not quite matured and he is edging on 160lbs) and they were bred for homestead protection. Google them and it will tell you that they are the most protective, non aggressive breed. From a puppy he has instinctively done this thing where, when any strangers come around, he will literally sit right beside you and lean into you so that he can keep a watch on the stranger while still maintaining physical contact with you. It's pretty cool. BUT, he likes to chase. Goats, deer, our other dog, the cat, the chickens. When he was a puppy, the first time he caught a chicken, I think he was unsure what to do with it and we were thankfully there to make him let it go unharmed. To be honest, we have him too much credit in the beginning. He would be fine so we would turn out attention elsewhere, then look back up to realize that he was chasing and - eventually eating a chicken. He killed and ate 5 hens from the established flock that was given to us when we started out with chickens before I just started leaving the birds penned all the time. For months we played musical animals, letting each of them out at separate times during the day - never together. It was a total pain. Then we eventually moved our chicken pen to the other end of our pasture, much further away from the house. We started letting the chickens out more and more, while letting Timba out at the same time (with me keeping eagle eyes on him). He would (and honestly still does) periodically try to chase. But he has not deliberately caught and killed one in nearly a year now. He has - during brief chases - swatted at and struck one hen and one of our ducks. He is so dang big that it cause both of them to have internal injuries that hey died from...
All this to say, I still free range my birds. We live on a large piece of property and the goats and birds are free to roam where they want - though they mostly only roam within 1-2 acres, maybe 3 from their pens. Timba is allowed out with them during the day so long as one of us is home. (My husband's a high school football coach, so he's home a good bit right now.) if we are gone, Timba stays in our shop. Thankfully, we are so active outside that we have not had any instances of day time predators (no coyotes, fox, etc,), knock on wood. Our predation problems have been at night, so now we do make both dogs sleep outside at night. The birds are penned, of course.
Ours is not a perfect free range scenario. But, again, the husband is home a good portion of the day. Thankfully, we live very close to the school so he comes home during the day during the schoolyear to check on everybody. I think (hope) that as he continues to mature that we can eventually trust him to stay outside all the time to guard the place, including the chickens. I'm continuing to work very hard on training him to understand that the birds are MINE and that chasing them is not okay. I've recently created a new pen for the juveniles that is right off our back porch. The juvies get to free range during the day as well since they stay right by the house. He has gotten MUCH better and will now mostly stay still, watching them if they come up to us while we are outside. I don't 100% trust him yet, but we're getting there.
From my understanding of dachshunds, they were initially bred for a very high prey drive (chasing and killing weasels, etc.), so I am not sure that you can train them otherwise. The pit bull mix I bet you could. Especially as he matures and if you train him to understand that the birds are off limits to him. I will say that I would maybe take him outside with the birds by himself (without he dachshunds). One thing we found with Timba was that of our kids were out (my kids also get a kick out of chasing and trying to catch the chickens), then Timba would get excited and try to join in the fun. If your dachshunds stir up havoc, then the pit mix will feed off of them and will never learn to be settled when he's out with the birds.
Sorry for the long post, and I'm not re-reading it before posting it - so please forgive any weirdness that my phone's auto correct may have done.