What leapt out at me from your post, was this
DP has a short fuse and stresses easily
DP says his mental health is collapsing ( and mentions work is also stressful)
DP "I can't do this he's screaming and hitting the food away. "
" I try and calm DP down as he's shouting and slamming doors'.
The baby (under dressed) had not been fed or given a drink, fell asleep in the highchair. (which probably ,means he screamed himself to sleep).
RED FLAG. What matters here, is that your DH is dangerously close to breaking point and the awful neglect is a clear indication the baby is at risk
Note, he describes the baby as uncontrollable, screaming and hitting (food away). My bet; DP's actually describing himself. His nerves are at screaming pitch, he's at the point of hitting out, he's losing control. And as soon as you get in, that's exactly what happens, he loses control, shouting and slamming doors.
Your baby was in real danger .His father repeatedly , urgently sent the urgent message he can't cope, he's at breaking point, come home, help. Those are the circumstances in which struggling, desperate , stressed-out parents snap, lose control; and hit, violently shake or throw a child to the ground. With tragic results.
All you said about DP';s crap parenting is true; of course you are angry resentful and taken advantage of; you want to work nursery is expensive. Its a mess and its not fair. But none of that has any importance right now, compared with keeping your baby safe
DP's right; he can't do it. Please, please believe him.