I think you need to keep repeating your assertion that you do meet Support Group criteria and supply as much evidence as you can. If the reconsideration does not find in your favour, and your case procedes to Tribunal,you will be sent the 'bundle' of paperswhich should contain everything used to make the decision.
Meanwhile, put your request for the Statement of Reasons in writing, or submit a 'Subject Access Request'.
Nothing on this board constitutes legal advice - always consult a professional about specific problems