Thank you for confirming.
Whilst he has no legal interest in the property given that it is in your name only, he has what is known as an equitable interest and you clearly acknowledge he has contributed equally to the property. Though an equitable interest does not give him the right to live there necessarily (in that he may have to apply to court for this right to be recognised given that the house is in your name) it does give him the right to his share of the proceeds/ equity.
For example, if you attended the property and asked him to leave, and he refused, you could call the police - it’s your house and is in your name. I imagine they would ask him to leave.
You could consider an occupation order as a last resort, though there has been no domestic abuse, there are competing rights of occupation at stake here and your son’s needs are combined with yours. The only weakness in this approach is that you have a temporary alternative in your parent’s home. If he has no alternative then this goes in his favour. Occupation orders are also only ever short term measures, you need the long term solution which is hopefully to buy him out.
If you intend to buy him out, then you can put your offer in writing inviting him to accept this and give up possession of the property to avoid a costly court case for which he will be held liable for the costs of, given that you have made a fair offer to buy him out.
There’s no formal template for this sort of letter, but if you search for a pre-action protocol letter you can base it loosely on that template/ format.
Is there anything more I can clarify for you?