Op l don't want to upset you as l can see you have a lot on your plate, l agree with others that your poor puppy needs rehoming fast.
BUT l am really horrified at your acceptance of your 4 yr olds cruelty, a dachound puppy is tiny at 4 months, really tiny, and he HURTS him!
I have a 7 month old cocker and 2 dcs, they were brought up with an adult Weimaraner as babies and toddlers, so l KNOW it is perfectly possible for Dcs of 4 to understand it is not acceptable to hurt a defenceless puppy or any animal.
The first time my dd grabbed my Weimaraners floppy ears and pulled, as a toddler l whipped her hand away,(very firmly) held her in front of me to ensure l had her attention, and explained in no uncertain terms how much that would hurt our dog and she was NEVER to do it again.
Inevitable something similar happened with my ds ( a year younger) at some point, and he got treated the same, they were far to worried about my reaction to ever hurt a pet again...(l hasten to add l do not smack or yell)
When you said he stood on her tail l could cry for the pup, and if that was my child l would have pushed him off hard and stood on his foot.
You are responsible for the safety of a small breed puppy, please get a grip on disciplining your Ds.