HUGE HUGE hun.
I don't know that I will have much useful advice, but I will give you some of my experience/observations. I am a childcare worker. I worked with a little boy, D (who was 3 when I first started with him) who as diagnosed with a variety of anxiety and OCD issues. When I first started at this centre I was not aware he had issues with anything. Sure, he seemed a bit hypervigilant with things like his bag and people/children sitting too close, he stood back from play with his peers but would sit and play with things with me or another carer, he was very touchy-feely with myself and the other carer (I later found out because he liked us and felt safe with us, so wanted to be close, though to the point of invading our 'space'). There were other things I can't think of now, but he had some very similar issues as your son with anxiety.

D was very settled int he environment of our centre when I started there. He had been attending 3days a week for about 8mths by then. When I later read his personal file and doctors reports, I got the bigger picture. He had been seeing a therapist for a while, his father had severe anxiety and depression. His mum was overly protective because she wanted him to be happy and he was mostly happy with her. But now he was happy with us too. After lots of work and nurturing he was able to happily say goodbye to his mum and then tell her about his great day when he went home. Before he left last year, he had made a niche for himslef with 3 of his peers. He interacted with the group relatively well, he was able sit in the group at story time, have children sit close to him even if they weren't his 'best' friends. He could be away from me and the other carer for the majority of the time.
What I am trying to say is that with some help and time, this little boy was dealing with his anxiety. He was having fun, he had friends, he looked forward to coming to childcare.

I think YOU are the best judge of your son. You and your DH. Not your MIL. She may know him, but she doesn't know him inside out like you do. You also know the signs of anxiety. I think you know what is the right thing to do. And you know what even if you do take him to a therapist of some sort, they can tell you yes or no. Yes he needs more help from a professional, or no he doesn't need help, you're doing a great job! As you said these people are professionals, they won't make matters worse and they can ease your worries, either way.