You can grant a term of less than 6 months but do be aware of the tenant's rights of security of tenure. Any order for possession will not take effect earlier than 6 months after the beginning of the AST. Therefore you can grant a periodic weekly tenancy. It will qualify as an AST but you cannot obtain possession until six months have passed from the commencement of the original tenancy.
Do you live at the property?
I agree, the existing arrangement will not work as the person is no longer a lodger.
You could seek to terminate the lodger's agreement. However, it is now a grey area as to what rights the tenant has now that the landlord has moved out. Check the terms of the lodgers agreement and seek to terminate.