You probably can't think straight as you are so tired. I have long term insomnia and I think this is due to having sleepless DC and DGC it ruined my sleep pattern forever.
What is your objection to going to the GP if you are so desperate that you would consider foster care without your DP why would you not discuss this with someone and seek help?
Also what is your objection to shifts can you sleep separately and wear proper ear plugs? Alternatively actually sleep somewhere else where you cannot hear him?
After trying everything with one of my DC we decided on that the only way we could function was to do shifts which meant one of us slept in the other room with ear plugs and the door shut. We didn't even try to put him in his cot, he slept with whoever's turn it was that night in our bed.
We also had the odd night in a hotel or at a friends both separately and together, we were lucky to have someone to stay with the DCs and do a night shift. We were so tired we slept and when you know you wont hear him its amazing how you tune out and sleep.
Eventually the GP prescribed medication and we co slept with an extra bed on the side, we had 7 foot of bed because he had to have physical contact and we needed to have some sleep.
Co sleeping was really frowned upon in those days and my HV was all for controlled crying. I was judged and told it was my lack of routine, (it wasn't) but I didn't care I just wanted sleep. I tried CIO and he cried until he vomited, there is no way I could have him that distressed, and it defeated the object.
He was almost 3 years old before he suddenly settled in his own bed.