I haven't had to ask for a copy of such a recent document, so I could be wrong.

I'm not sure why you think that you need to have the BMD reference number. If you order direct from GRO then I think that you have enough information to proceed.
From the GRO website -
Q3. What information do I need to apply online?
To get the fastest response, you should provide the GRO index reference number for the certificate that you are interested in, but it isn’t essential. You will be prompted to supply further details for more recent births and deaths (within the last 50 years) as you complete the online application form. If you are unable to complete the mandatory fields, then you will not be able to apply online.
Q4. If I don’t have the index reference number what will happen?
If you do not have the GRO index reference you may still apply online for any event which took place on or after 1st July 1837 to 6 months from the current date (for marriages this period is extended to 18 months) provided you have sufficient information to identify the entry. A 3 year search for the index reference will be carried out. You should supply the exact date of event if you have it. If you don't have the exact date, enter 01/01/YYYY and we will search the specified year and one year either side. If you do not provide an index reference number, your application will take up to 15 working days to process. There is a separate policy for events which have taken place in the last 50 years and full identifying information must be provided (see Q7 under “Important information on delivery times and policies.”
Q7 just states that you cannot order certificates for events within the last 6 months (extended period for marriages).