You have no reason to go on the mortgage, your share is paid for and you are not borrowing any money, your brother is. He will need to get a mortgage in his sole name to cover the cost of the share he is buying from you.
You can sell half your home to your brother. Do it as tenants in common instead of joint owners and then you can will or dispose of your half to whoever you wish. It is a very sensible option for married couples to purchase property as tenants in common rather than joint owners because your half would automatically go to the other joint owner on your death, and you wouldn't even have the option of willing your half to whoever you may wish