Hi and welcome to beyond blue. It definitely sounds like you are in a tough space.
I have 2 teens. My daughter is a bit of a perfectionist. However she does not give up. Good, but can drive you crazy sometimes.
I am not saying that your son does not have social anxiety, but rather the perfectionist traits are affecting other aspects of his life. It is an all or nothing tendency. A cognitive disorder/distortion. And it sounds like it is reinforced by the responses he gets from his father. Maybe trying to get some praise from his dad.
On the support side of things... Maybe have positive statements on post-it notes. Maybe put them into lunch box or pencil case. Or around home. Can you get him to say the statements out loud. For example "all I can do is my best". Repeat this enough times and you might start to believe it.
I have an app called virtual hope box, recommended by my psych for different reasons. One of the distraction tools is a word finder puzzle. Thing is, it only contains positive words. Eventually you start to believe what you read.
Could you also introduce him to famous people that made mistakes. For example the number of attempts Edison took to create the light bulb.
There are also quotes that talk about success through failure.
On the psych sessions... Unfortunately it is not a sprint to the finish line. It takes time, and sometimes it will be 2 steps forward and 3 back. And that might also be viewed as failure. That is from one perfectionist to another.