Does he cry for everyone, all the time? Was he crying on the call when your DH was holding him on the call?
I'd have left him with DH if he was calm, trusting that DH could mute himself if baby got upset until I could go get him. I WFH and don't have kids, but have often held friends' babies on calls to help them. It's not ideal, but it's not like concentration is much better with a screaming baby in the background!
I've managed the baby room of a nursery, where a lot of parents would be stunned that their baby would calm down when they left. We'd let some sneak into the office before pick-up time and see that they really were happy and calm. I've also been a mothers help, where technically I wasn't supposed to have the baby alone, and a nanny for newborn twins, one of whom cried a lot, but that I loved! My charges were great.
If you need help, choose how you'd prefer to get it. You could settle him in a nursery for a few hours a week, or get a mothers help or nanny, although you might find that it's tough to get just a few hours a week there as they will want to find families to fit around you.
Then you can relax the pressure on your DM and MIL. It's a shame that they aren't offering you the same care that they got, or that you expected, but they seem to have a lot of other caring responsibilities, rightly or wrongly. They also don't sound keen to look after him. You can either ask them outright, which you don't seem to want to do, or just try to lessen the resentment by getting your help elsewhere.
I'd keep pushing for an answer as to why he's so sad, too. In my experience, babies are often called "high needs" until the problem becomes apparent, and then they are totally different little things.
All the best 