Hi there - I feel for you in this transition time.
I have twin boys who slept in cots in the same room. When they began climbing out of their cots, I knew it was time for beds, but didn't immediately get them out of the cot.
I erected the beds and put them into the same room as the cots, but made sure I put the boys to bed in their cots - just so that they got acquainted with having the beds in their room. The beds were made, and ready for use, and I told the boys that they were special beds and that only big, grown up, special boys were allowed to sleep in them.
After about a month of having the beds in their room, I removed the cots and made a great big thing of "tonight boys, you get to sleep in the special, big boy, beds".
Never really looked back after that. They loved them. They loved the fact that they could climb in and out and, although we did have a couple of nights of them getting up at all hours and coming through to mummy and daddy's room, they now sleep straight through unless they need the toilet.
Perhaps what is bothering your son is the gate on the door. I know my two absolutely hated being behind a gate. Is there any way to move the gate to the top of the stairs instead of his room? We did this, so that they still could get up and come in to us if need be.
We use a nightlight and leave the door ajar at night, so they know they can get out.
Hope your situation improves soon.