First and foremost: When a child of that age cries, then that means the child is distressed. The reason we usually cannot let a baby cry is that, over millions of years, crying has become a statement of "something isn't right!" and hearing a baby cry has become a reminder that "something needs to be made right!" It's built-in – both into the baby and the parents. So do not let others talk you into "some crying isn't bad, wait it out." A baby cries when something is wrong, and our natural reaction is to try to remove whatever makes the baby feel bad. What worked fine millions of years shouldn't be easily dismissed for some cultural habits gained a few dozen years ago.
Working from that, the question becomes "What is wrong and what can we do about it?"
What is wrong I could only speculate. One thing immediately coming to my mind is that, for millions of years, a child that didn't feel contact with a familiar adult had been lost in the forest and needed to draw attention so the parents can pick it up. This, too, is built-in. IME, some babies need more physical contact and others do less. But physical contact usually helps make the baby feel safe and peaceful again. That is, why we feel the urge to pick up a crying baby, after all – this was the right thing to do for millions of years, and it still works.
So my suggestion is to make it easier for you and the baby's mother to make the child feel good, so that the two of you will also get enough sleep to get through that period. (Always remember the parent's mantra: "It's just a phase. It will pass.")
One thing we have done was to build a small extension to our bed where a child can easily be put to sleep alone, but just as easy be picked up from when it cries. It was about an inch lower than our mattress, so that the children would not roll into our bed by themselves. (Some children "travel" a lot during sleep.) Putting it on my side of the bed somewhat lowered the lure of wanting to be fed in the middle of the night. I would lie on my back, the baby on my belly until the baby was quiet and comfortable. Then I'd slowly roll to the side and let the baby slip to its "balcony". If the baby starts to cry, I'd pick it up again. After a few nights this could become routine for the two of you and the child might be more relaxed about lying alone.
One thing you also might want to try is tight-wrapping the child, which helped with one of my kids.