I think that the Treasurer is concerned whether you should be self employed and if you undertake administrative work then I feel you should be employed for this specific work load.
But no you are not duty or legally bound to include your UTR on any invoices the only infromation they must contain as detailed by HMRC are
Invoices - what they must include
You must clearly display the word ‘invoice’ on the document. You must include:
- a unique identification number
- your company name, address and contact information
- the company name and address of the customer you’re invoicing
- a clear description of what you’re charging for
- the date the goods or service were provided (supply date)
- the date of the invoice
- the amount(s) being charged
- VAT amount if applicable
- the total amount owed
Sole trader invoices
If you’re a sole trader, the invoice must also include:
- your name and any business name being used
- an address where any legal documents can be delivered to you if you are using a business name
Limited company invoices
If your company is a limited company, you must include the full company name as it appears on the certificate of incorporation.
If you decide to put names of your directors on your invoices, you must include the names of all directors.
You must use VAT invoices if you and your customer are VAT registered.
These include more information than non-VAT invoices.
Let me know if I can assist further